Usually they just sleep. Roxy is an unusual case. However, sometimes they can be awake without realizing it like Dave who just hung out in his drem room makin beats.

They have Conditional Immortality. They can die, but only if their death qualifies as Heroic or Just

Matesprit is a romantic partner whom you love and can mate with. Very similar to human romance.

Kismesis is an intense rival that you compete with and can also mate with. Explained more in-depth here by Karkat. 

Moirail is a platonic soul mate that keeps you emotionally balanced.

Auspistice is a troll who prevents two other trolls from entering a kismesisstude by playing peace keeper and mediating between them whether for personal reasons or because they think the kismesisstude could upset others. 

Only if they wish for these injuries to be healed. For example, if the prosthetic arm was something he liked about himself or felt pride in then it won’t heal during godtier. But if it’s something he didn’t want, like Vriska and her metal arm then it will heal. 

They can’t, but those that are god tier can change into their costumes at will

That would most likely be the current alive Gamzee, either dreaming or physically passing through the bubble

There is

Their pre-scratch selves are clones of themselves. It’s a paradox, they needed to exist so that their ecto-babies could be created so that they could exist.

Hussie in the author’s notes of Homestuck book 3:

It never even comes close to being explicated in the story, but it is strongly implied this is where Rose’s mom grew up. Though after the introduction of Roxy, and another reread, this probably becomes more obvious. Who raised her? Was it the owner of Skaianet, a younger grandpa Harley? Did he outsource parenting duties to someone else in the lab? Maybe some exiles? If so, she had an upbringing fairly similar to that of Roxy in the future. These mysteries have LEGS