Hey, check out "Oxford Press" the right sentences in written English are: 1) DOING the shopping, we went back home. 2) HAVING DONE the shopping,we went back home. 3)AFTER DOING shopping, we went back home. it is NOT right "after having done shopping". RULES: a) When there are 2 short actions, one straight after the other, we can use an -ing form for the first action. Opening the bottle, Mark poured the drinks (= he opened the bottle and then poured the drinks). But we can use the perfect -ing form as well. Having opened the bottle, Mark poured the drinks (= after opening the bottle, Mark poured the drinks). b) IF EITHER OF THE ACTIONS IS LONG, WE MUST USE THE PERFECT. HAVING PHOTOCOPIED all the papers, she put them back in the file. The structures "1", "2", "3" are typical of written English. In SPOKEN ENGLISH , to talk about one action after another we use a sentence like this: I opened the bottle and then poured the drinks. In conclusion, in your own situation, the best forms are the number "2" and "3". I hope it helps.

"After having done it" means the same as "Having done it". "After" is a superfluous word. There's a small difference I can see between "after doing it" and "having done it". Consider the sentences (1) "After doing the shopping, it started raining" (2) "Having done the shopping, it started raining" Neither is wrong, but (1) is much better than (2). The problem with (2) is that the fact that it started raining has NOTHING to do with the fact that the shopping was done - it's just coincidence. But the sentence makes it sound like it started raining BECAUSE the shopping was done, which is nonsense. But this is a subtle point, nobody would notice, and it's not really an error. But stylistically, (1) is better.

Nothing really. Both are finished actions. The second one is said when the action is still relevant to the present moment or the conversation in some way.