What you want to do

싶다 is a helper verb: it means 'wanting' but cannot take an object or stand on its own, so it leans on the verb before it. That verb gives up its -아요/-어요 ending and takes 고, which never changes shape, so 먹고, 가고 and 마시고 need no vowel rule at all. The vowel rule from day 13 still applies, but only once, at the very end: 싶 has the vowel ㅣ, so it takes 어요, giving 싶어요. Everything the sentence needs, tense and politeness included, is carried by 싶어요.

Asking about someone else's wishes with 뭐 하고 싶어요? is fine, because you are inviting them to speak for themselves. Stating another person's wishes is different: Korean treats wanting as an inner feeling that only the person who has it can report directly, so 'my friend wants to go' needs another form, which comes later in the course. For now, -고 싶어요 is for 'I' in statements and 'you' in questions. To negate it, 싶다 behaves like any other verb and takes the -지 않아요 of day 21: 싶지 않아요.

English 'want' happily takes a noun, 'I want water', but 싶다 only attaches to verbs, so Korean usually says what you want to do with the thing: 먹고 싶어요, 마시고 싶어요. There is a verb 원해요 'I want', but it is heavy and formal and sounds odd over a café counter. For things you need, use 이/가 필요해요; for things you would like to own, 갖고 싶어요 (gatgo sipeoyo) 'I want to have'; and when you are asking someone to hand it over, 주세요. 보고 싶어요 means 'I miss you' because Korean expresses missing someone as wanting to see them, which is the more concrete of the two ideas.