How do you judge the strength of an inductive argument, and can a strong one have a false premise?

Ask the core question: "if the premises were true, would the conclusion probably be true too?" Strength is about that link — so yes, a strong argument can still have a false premise.

The core test for inductive strength: "If the premises were true, would the conclusion then probably be true?" Strength is about the premise→conclusion link, independent of whether the premises actually hold. So a strong inductive argument can rest on a false premise and still have a likely-true conclusion:

(P1) All previous US presidents have flown on Air Force One. (K) So the next president will probably fly on Air Force One.

(The premise as stated isn't even strictly true historically, yet the conclusion is reasonable.)

Tip: This mirrors the deductive validity/soundness split. "Valid/strong" is about the connection; whether the premises are true is a separate question. Judge the link first, the inputs second.