The title pretty much sums it up. Is the President not immune to being sued in cases related to his agencies' performance?
First, who is starting the lawsuit (as long as they have standing) does not make a difference.
Sovereign immunity means that the government (including public officials) cannot be sued unless the government consents to it. The government includes all of the public officers while acting in their functions.
The basic of it is the separation of powers: if the executive power may be sued, it could lead to the judiciary power stepping into the functions of the executive power.
This what somewhat loosened by the Federal Tort Claim Acts, that allows the state to be sued due to tort acts. But even that does not allow to sue for underperfomance (from the article,
The FTCA exempts, among other things, claims based upon the performance, or failure to perform a "discretionary function or duty").
Of course, the alternative for Congress is to impeach the president, but that is a political process.
You, personally, cannot sue the President or any member of Congress for any act they have done in the service of their office. You can however, sue the government as a whole for some legislation they have passed or some action an executive agency has committed, so long as you have standing to do so. You can also sue the President or a member of Congress individually for something they did that was not connected to their official duties, but it can't have anything to do with their job as a public official. A possible example of this would be if a senator punched a protester at a political rally; that person could sue them for assault
Congress, however, CAN sue the President (and they have done so many times), but the President cannot sue Congress. This is because in the American system, the President is supposed to act according to the laws Congress passes. They can hold him responsible for failing to do so, or acting where he did not have authority. But the President does not participate in the legislative process nor direct the actions of Congress (except to veto bills), so he would never have standing to sue them.