Learn English – it’s been two years… + present perfect or simple past

Both the sententence are grammatically correct.

The word until is used as a preposition and conjunction

When you use it as a conjunction, it means "up to the time that". You usually use the present simple in the until-clause, but it's possible to use the present perfect, without any change in meaning. If there is any subtle difference, that is in regard to the degree of emphasis.

Your second sentence with the present perfect in the until-clause puts more emphasis on the completion of the action in the until-clause.

The OP is right that both the present simple and the present perfect in the until-clause are indicative of actions or events in the future.

Your first sentences is correct

They seem not to have learned grammar.

Your second sentence needs to be restated from

They seem haven't learned grammar

to

It seems they haven't learned grammar.

Your third sentence is almost correct, the contraction needs to be expanded because of the "to"

They seem to have not learned grammar.