Yes, 'I remember going swimming when I was in Ibiza' is fine. There isn't a problem with having two 'ing' forms one after the other. The activity is 'go swimming' ( the same as 'go cycling', 'go skiing', 'go sailing' and dozens of other activities). The word 'swimming' never changes. Only the verb 'go' changes. You need to have the gerund of 'go' following 'remember', so the correct sentence is 'I remember going swimming'.

Yes, it's perfectly correct. I'm in a (very) amateur production of a musical entitled "Once Upon a Mattress," and we're learning a song which happens to contain this line: "I'm going fishing for a mate..." (It's sung by Princess Winnifred, who is looking for someone to marry.) I might say to my wife "I'm going shopping for groceries."

They are both correct.