Learn Portuguese (Portugal): Não parece que o museu exiba uma falsificação. - It does not seem that the museum exhibits a forgery.

The phrase Não parece que (It doesn't seem that) expresses doubt or unlikeliness, which triggers the subjunctive mood for the following verb. If the sentence were affirmative (Parece que...), it would express a strong impression of reality, and you would use the indicative exibe.

What happens if we move the não to the second clause, like Parece que o museu não...?

The grammar and meaning both change. Parece que o museu não exibe uma falsificação means 'It seems the museum does not exhibit a forgery.' Because the main clause (Parece que) is now affirmative, it states a strong belief or probability, so the second verb switches back to the indicative (exibe).

How is the subjunctive form exiba created from the infinitive?

The verb is exibir (to exhibit or display). As a B2 refresher on forming the present subjunctive for regular -er and -ir verbs: you take the eu form of the present indicative (exibo), drop the -o, and add the appropriate subjunctive ending. Here we use -a for the third-person singular (o museu).

Could I use the word falsa instead of falsificação?