If someone sues President Trump for slander, do the taxpayers fund his defense? I would consider having the Attorney General as counsel would qualify.
Also, would the taxpayers be on the hook for any settlement of a lawsuit against the President?
A president can be personally sued, and does not enjoy universal immunity while in office, see Clinton v. Jones, 520 US 681 – in that case, Clinton was represented by private counsel. There are differences between that case and the instant hypothetical, the most prominent being whether such statements might be shielded because of executive privilege. The primary procedural question would be whether a potentially defamatory statement was made in connection with official duties. The limits of executive privilege are not at all clear. That kind of constitutional issue could involve the solicitor general. Let us assume that SCOTUS finds (somehow) that a particular statement is completely outside the ambit of executive privilege (separation of powers). Then it is not an interest of the US government to defend the personal interests of a defendant even if he in in office (whereas it is an interest of the US goverment to defend the official interests of POTUS), and private counsel would represent the defendant. Any settlement or award would be out of the defendant's pocket (as was the case in Clinton v. Jones).