Yes, he gets a pet called Nuts or Na-tsu, which is obviously Tsuna backwards.
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In chapter 352(at the end of the chapter),Tsuna found out that Reborn wasn't actually a baby. In chapter 353, Tsuna is still thinking about Reborn's adult form.
Byakuran and Tsuna start battling in episode 200 and the fight ends in episode 202.
At this point in time, probably not, but when he gets stronger/older, probably. That's the things about main characters in a Shounen manga- no limits to their power.
she calls him maggot most of the time
Yes, there is likely to be Ketekyo Hitman reborn episode 205 in animation later this year.