Yes and No. The Federal U.S. Government as well as each individual state and territory claim Soverign Immunity to a limited degree. In the case of Federal Government, they claim Soverign Immunity to a near limitless degree (more on this in a moment), while States and Territories claim "State Soverign Immunity" which grants them immunity with respect to all soverign powers granted under the U.S. Constitution.
To clarify, each U.S. State is seperately soverign in all matters not explicitly allocated to the Federal Government by the U.S. Constitution and all constitutionally written laws. Where the Constitution does not in writing claim a power for the federal government nor reserver a power for the state government, the power is assumed to be state government. So a State can claim immunity for a police situation but not a law it makes regarding immigration or foriegn policy, since that is the exclusive wheelhouse of the Feds. Per the 11th amendment, states can only be sued in state court by citizens of the state, while non-state citizens wishing to sue a state must sue in the Federal Court (Which for the purposes of the trial may adopt the defendant State's laws and regulations for the purposes of questions of fact and law brought up in the suit).
All U.S. States and the Federal Goverment additionally wave Soverign Immunity under specific conditions outlined in laws. In the case of the Federal Government, the laws governing when this immunity is waived are governed by the Federal Torts Claims Act (1946) and the Tucker Act (1887). The FTCA covers intentional torts by the Federal Government, while the Tucker Act typically covers matters related to contractual obligations or monitary interactions with the federal government. Things that may be sued for include but are not limited too:
- Intentional Torts committed by Government Agents in their Official Capacity (i.e. When the FBI violates your rights or maliciously investigates you).
- Contractual disputes where the Government is a party to the Contract, either in written or implied terms.
- Constitutional Claims (i.e. The Government may have violated the Constitution. These tend to be for non-monitary relief such as mandemuses, though the 5th Amendment's Taking Clause can be invoked of the government took property and does not justly compensate.).
- Individual Agencies of the Government when the monitary claim is not drawn from the U.S. Treasury (the agencies own allocated budget is used to pay).
- Refund for taxes paid
In all cases, a Judge will determin if the Government, under the laws, can be sued. If in the example of a government employee being sued, the Justice Department will first make a determination if the employee was acting as an agent of the government when they committed the tortious act, and then will declare themselves Respondent Superior through the agency of employement. From there, the case becomes a suit against the Federal Government and FTCA immunity waiver is determined by a Judge.
For example, lets say you (Mr. Rapt) sue Special Agent Jimmy Jones of an intentional tort that he committed in the course of his duty as an FBI agent. The DOJ will first declare that they are Respondent Superior in the case of Rapt v. Jones, thus turning the case to Rapt v. United States Government, Department of Justice, Federal Beura of Investigations, and Jones (Rapt v. United States, et. al.). From here, the Judge who gets the case then determins if the U.S. Government under FTCA can be sued by Rapt, based on Rapt's citizenship status (usually permanent legal residents are covered as citizens for this point, but I can say for certain in this particular law), ripeness (the suit must be filed within 2 years of the incident or first knowledge), and immunity waiver under FTCA.
In the case of the State Government, they should have similar laws, but I'm not going to identify all 50 individual laws here. Additionally, the 11th Amendment permits U.S. Congress to waive a State's immunity to suit in certain cases (mostly comes up with the Due Process Clause in the 14th Amendment.).
As a fun point, in all cases where a government appears as a defendant in a suit, it is automatically a Bench Trial, as the government usually include a waiver of their right as defendant to a jury trial. The thinking here is that, as a country that was founded by rebelling against the government, a jury of 12 of the government's citizens aren't going to give them any slack at trial, where as a judge (who they pay) is much more likely.
TL;DR: Both questions are yes. The immunity is waived under federal law for certain cases, but it is a judge who will make the ruling if the specific case meets the aformentioned waived immunity.