Is it correct to say "proud of having met you", or should it be "proud to have met"?

"I have proud of having met both of you" is incorrect for two reasons. First, "have" lacks a valid complement. Second, the adjective "proud" does not seem to have any valid function.

Your next two examples (numbered 1 and 2) are both grammatically correct. Both use the perfect aspect ("to have" + past participle), so both indicate that the action happened before the present time.

In the first, the gerund "having" is the object of the preposition "of". In the second, the infinitive "to have" is a dependent of the adjective "proud". You might see it described as an "adverb", "infinitive complement", etc.

The verb "meet" has several meanings, but if you mean that the people were introduced for the first time, then the action can not be ongoing. It is finished.

Both are correct, but the second sounds more natural, at least in American English.

There is a very slight difference in meaning between the two sentences. The first sentence emphasizes the speaker's pride, whereas the second sentence emphasizes the act of meeting the listeners. The second sentence therefore sounds slightly more humble.