Others have addressed the power of a state court to subpoena a President (correctly IMHO), subject to privileges for national security secrets and confidential advice regarding a President's official duties.
Pending actions against the President for defamation outside an official capacity action have not dismissed those cases as a matter of Presidential immunity, nor have civil lawsuits alleging violations of the emoluments clause been dismissed as a matter of Presidential immunity (standing is a trickier issue in those cases). A trial in a civil case (which is not subject to speedy trial mandates) might be deferred until after a President leaves office in many cases without dismissing the case.
I think that there is little doubt that a dissolution of marriage action could proceed while a President was in office (this has happened several times in the cases of state governors who have analogous immunities in these circumstances, and in the case of foreign heads of state which are fully analogous but have different legal systems), or an action to establish paternity for a non-marital child (something that the British P.M. could have faced if there had been a dispute).
I think that there is little doubt that there are no special exemptions apply to a President who is a creditor on account of a personal debt in a probate case, or an heir in a probate case, or in a lawsuit arising from personal co-ownership of real property (e.g. a partition action of a family owned ranch owned by several co-owners).
I think there is a credible argument that the 25th Amendment to the U.S. Constitution pre-empts the authority of someone to bring a civil proceeding against the President to subject the President to a guardianship or conservatorship proceeding on the grounds that he (or she) lacks capacity while serving as President when no acting President is serving in his stead.
The impeachment language of the U.S. Constitution implies that a President may be prosecuted for at least some crimes committed while in office, in Article I, Section 3 of the U.S. Constitution which states that:
Judgment in Cases of Impeachment shall not extend further than to
removal from Office, and disqualification to hold and enjoy any Office
of honor, Trust or Profit under the United States: but the Party
convicted shall nevertheless be liable and subject to Indictment,
Trial, Judgment and Punishment, according to Law.
This clause sheds light on substantive immunity but not on immunity from prosecution or lack thereof prior to an impeachment conviction.
There are some widely recognizes Presidential immunities from criminal and/or civil liability. Most importantly, there is absolute immunity from civil and criminal liability at any time (in office or afterwards) for the President's conduct of his official discretionary duties in an official capacity (judges, prosecutors and legislators have comparable immunities).
There might be some exceptions to this for violations of federal laws targeted at official capacity conduct (e.g. bribery, theft of government property, Posse Comitatus Act violations, etc.) if the prosecution is for official conduct after being removed from office. But the Supremacy Clause of the U.S. Constitution probably bars criminal prosecutions for the official misconduct of a President while in office under state law.
There is not a widely recognized immunity from state law crimes committed by a President while in official for unofficial conduct.
For example, there is certainly not a consensus, and probably not even a majority of scholarly opinion that would view a President as being immune from a state law arrest or criminal prosecution for killing or grievously assaulting his wife or abusing his child in a hotel room after a campaign rally within the territory in a U.S. state. But procedurally, a President would be entitled to every matter of deference that does not actually abrogate a legal requirement (e.g. leave to testify by telephone, cooperation in scheduling hearings within speedy trial limitations, affordable bond pending trial or personal recognizance bonds).
Some of the opinions of the U.S. Supreme Court in its recent decision this year in Trump v. Vance on Presidential immunity from a criminal subpoena imply that there is not immunity, at least, for prosecution for crimes committed by a President before taking office (necessary to avoid a statute of limitations issue, for example), although there might be strong grounds to defer a trial in most or all cases.
From the official syllabus of the case (citations omitted):
Here, the President claims that the Supremacy Clause gives a sitting
President absolute immunity from state criminal subpoenas because
compliancemwith such subpoenas would categorically impair the
performance of his Article II functions. The Solicitor General,
arguing on behalf of the United States, claims that a state grand jury
subpoena for a sitting President’s personal records must, at the very
least, meet a heightened standard of need.
The President’s unique duties as head of the Executive Branch come
with protections that safeguard his ability to perform his vital
functions. The Constitution also guarantees “the entire independence
of the General Government from any control by the respective States.”
Farmers and Mechanics Sav. Bank of Minneapolis v. Minnesota.
Marshall’s ruling in Burr, entrenched by 200 years of practice and
this Court’s decision in Nixon, confirms that federal criminal
subpoenas do not “rise to the level of constitutionally forbidden
impairment of the Executive’s ability to perform its constitutionally
mandated functions.” Clinton v. Jones. But the President claims that
state criminal subpoenas necessarily pose a unique threat of
impairment and thus require absolute immunity. His categorical
argument focuses on three burdens: diversion, stigma, and harassment.
The President contends that complying with state criminal subpoenas
would necessarily distract the Chief Executive from his duties. He
grounds that concern on Nixon v. Fitzgerald, which recognized a
President’s “absolute immunity from damages liability predicated on
his official acts.” But, contrary to the President’s suggestion, that
case did not hold that distraction was sufficient to confer absolute
immunity. Indeed, the Court expressly rejected immunity based on
distraction alone 15 years later in Clinton v. Jones, when President
Clinton sought absolute immunity from civil liability for private
acts. As the Court explained, Fitzgerald’s “dominant concern” was not
mere distraction but the distortion of the Executive’s “decisionmaking
process.” The prospect that a President may become “preoccupied by
pending litigation” did not ordinarily implicate constitutional
concerns. Two centuries of experience likewise confirm that a properly
tailored criminal subpoena will not normally hamper the performance of
a President’s constitutional duties.
The President claims this case is different. He believes that he is
under investigation and argues that the toll will necessarily be
heavier in that circumstance. But the President is not seeking
immunity from the diversion occasioned by the prospect of future
criminal liability. He concedes that he may be investigated while in
office. His objection is instead limited to the additional distraction
caused by the subpoena itself. That argument, however, runs up against
the 200 years of precedent establishing that Presidents, and their
official communications, are subject to judicial process, see Burr,
even when the President is under investigation, see Nixon, . . .
Finally, the President argues that subjecting Presidents to state
criminal subpoenas will make them “easily identifiable target[s]” for
harassment. Fitzgerald. The Court rejected a nearly identical argument
in Clinton, concluding that the risk posed by harassing civil
litigation was not “serious” because federal courts have the tools to
deter and dismiss vexatious lawsuits.Harassing state criminal
subpoenas could, under certain circumstances, threaten the
independence or effectiveness of the Executive. But here again the law
already seeks to protect against such abuse. First, grand juries are
prohibited from engaging in “arbitrary fishing expeditions” or
initiating investigations “out of malice or an intent to harass,”
United States v. R. Enterprises, Inc., and federal courts may
intervene in state proceedings that are motivated by or conducted in
bad faith. Second, because the Supremacy Clause prohibits state judges
and prosecutors from interfering with a President’s official duties,
any effort to manipulate a President’s policy decisions or to
retaliate against a President for official acts through issuance of a
subpoena would be an unconstitutional attempt to “influence” a
superior sovereign “exempt” from such obstacles, see McCulloch v.
Maryland. And federal law allows a President to challenge any such
allegedly unconstitutional influence in a federal forum.
A state grand jury subpoena seeking a President’s private papers need
not satisfy a heightened need standard, for three reasons. First,
although a President cannot be treated as an “ordinary individual”
when executive communications are sought, Burr teaches that, with
regard to private papers, a President stands in “nearly the same
situation with any other individual.” Second, there has been no
showing here that heightened protection against state subpoenas is
necessary for the Executive to fulfill his Article II functions.
Finally, absent a need to protect the Executive, the public interest
in fair and effective law enforcement cuts in favor of comprehensive
access to evidence. Rejecting a heightened need standard does not
leave Presidents without recourse. A President may avail himself of
the same protections available to every other citizen, including the
right to challenge the subpoena on any grounds permitted by state law,
which usually include bad faith and undue burden or breadth. When the
President invokes such protections, “[t]he high respect that is owed
to the office of the Chief Executive . . . should inform the conduct
of the entire proceeding, including the timing and scope of
discovery.” Clinton. In addition, a President can raise
subpoena-specific constitutional challenges in either a state or a
federal forum. As noted above, he can challenge the subpoena as an
attempt to influence the performance of his official duties, in
violation of the Supremacy Clause. And he can argue that compliance
with a particular subpoena would impede his constitutional duties.