Which conditional is this sentence " I'd take care of you if you'd ask me to"?

Yes, that works. In other sentences it's the same. Like this one, I'd hug you if you'd want me to.

"...if you'd ask me to" suggests "if you were kind/trusting enough to ask me" (but you seem unwilling to do that). Adding "only" or "just" underlines that subjectivity: "...if you'd only ask me." Other examples: "You'd understand what she's saying if you'd just stop interrupting"; "I could do a bit of gardening if it'd only stop raining", where "it" is not a human agent but is still capable of doing me a favour. This use of will/would and won't/wouldn't is seen also in non-conditionals, with the meaning of insistence or refusal, to the speaker's annoyance: "He will keep leaving the door open"; "They would come home drunk"; "The engine won't start"; "The door wouldn't lock properly". A heavy accent on "will" and "would"---without contraction---emphasises the annoyance! Caveat: I'm from the UK; my usage does not necessarily coincide with that of other native English speakers around the world.