The US President is indeed bound by the Constitution, and indeed by the ordinary laws. Current Justice Department policy is that a sitting president may not be indited. No court has ever held this, the US Constitution does not give explicit presidential immunity the way it gives limited immunity to members of congress (in the "speech and debate" clause). No sitting US President has ever been charged with a crime, much less indicted, so the matter has never come before a court.
An old news story indicated that President Grant was stopped for a traffic offense (speeding, in a horse-drawn carriage), accompanied the officer to a police station, paid an appearance bond for the traffic court, and then failed to appear, forfeiting the bond. Even if this is accurate, no claim of presidential immunity was made, and no court decision was rendered. So no precedent was established by that event, one way or the other.
Any President may be impeached and convicted, if Congress sees fit to do so. There is no enforceable standard on just what is and is not an impeachable offense. That is left to the sound judgement of Congress. Nor is Congress required to act if it chooses not to, no matter how strong the evidence may be.
Nixon's Vice President , Spiro Agnew, was investigated for alleged corrupt practices. It appeared that Maryland (where he had been Governor) was ready to indict him on several charges. He was persuaded to plead "no contest" in a plea bargain to a single count, and was sentenced to probation with no jail time. At the same time, he resigned as VP. No one knows what would have happened had he continued to insist on his innocence, and claimed before the court that a sitting VP could not lawfully be indicted (a claim he had made earlier in the process).
Even assuming that a sitting President cannot be indicted or tried, nothing prevents such a person from being charged and perhaps convicted after his or her term has ended. The constitution explicitly says that if an official is impeached and removed from office, there may be a subsequent trial on any relevant charges.
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. (Art I; section 3; clauses 6&7)
Whether any statute of limitations would be tolled (paused) while the president was in office cannot be determined until and unless the matters comes before a court for a decision.
Note that the US Constitution imposes few duties or obligations on individuals. it is mostly concerned with specifying he structure of the federal government, and the powers of and limitations on its various parts. It also specifies the relations between the Federal and State Governments. It also declares a number of rights held by individuals, most of which can be regarded as limitations on the power of the government. If the President were to be accused of a crime, it would almost surely be one established by statute, not by the constitution directly, because treason is the only crime defined directly by the constitution.
However, the official acts of the president are clearly limited by the Constitution, and in a number of cases have been held void as being unconstitutional. One of the more famous cases is Youngstown Sheet and Tube vs Sawyer 343 U.S. 579 (1952), also known and the "steel mills seizure case". During the Korean War, President Truman attempted to take control of a number of steel mills to stop a labor dispute, on the ground that this was hindering the national defense. The US Supreme Court ruled that he lacked the power to do this, and that his action was void.