Can HIV go into remission?

Not exactly. Hiv causes a chronic infection for which there is no current cure. With HIV medication patients can expereince a longer disease free life. However this period of improvement only exists as long as patients continue on treatment. A disease free period without medication does not exist for 99.5% of HIV patients. Close monitoring of blood work & response to HIV medicine is required throughout life.
No, not that we know. No, I'm not aware that HIV can go into remission (like after cancer treatment). Anti-HIV meds can prevent HIV from existing in the bloodstream, but HIV can be latent, which means that it may hide in the body (such as lymph nodes), but not be actively reproducing. In these locations, our anti-HIV meds cannot fight HIV in these locations. This is why we don't have a cure, as HIV can hide there.