They're both correct and the difference in meaning is minimal. Just be aware that if you use the conditional "I would love," then you need to follow it by the infinitive, not the gerund: "I'd love to do that."

Both are correct; they just suggest slightly different things.

No, the first one is not correct. "Loved" is past tense, "to go" is present. It would be correct to say: * I love to go to my aunt's house. (no specific time) * I loved going to my aunt's house. (very recent past tense, or a more distant past that is non-repeatable, such as "before she died", "when I still lived near her", or "when I was little".)