Learn Italian: Ho il presentimento che le aspettative siano troppo alte. - I have a premonition that the expectations are too high.

Usually, no. The optional omission of che before a subjunctive clause applies almost exclusively directly after verbs of thinking, hoping, or fearing (like credo, spero, or temo). Because this sentence uses a noun phrase (il presentimento) to trigger the clause, dropping che sounds unnatural. You should keep it here.

Why is 'siano' in the subjunctive instead of the indicative 'sono'?

The phrase avere il presentimento che (to have a premonition or feeling that) introduces a personal perception, doubt, or an unverified idea. Just like the verb credere (to believe), this kind of subjective noun phrase requires the subjunctive mood in the following clause.

The subject 'le aspettative' is feminine plural, so why is it 'troppo alte' instead of 'troppe alte'?

In this sentence, troppo is an adverb modifying the adjective alte, meaning 'too high'. Adverbs are invariable in Italian, so they never change their endings. You would only change the ending if troppo were an adjective modifying a noun directly, such as troppe aspettative (too many expectations).

Can 'aspettative' be used for physical waiting, like expectations for a train's arrival?