Hell yes. He’s much less grim, seems to have got over the death of the Time Lords to a much greater extent, and exhibits a constant upbeat zaniness. He’s very fun loving, willingly participating in jokes, but has also exhibited a much darker core than even Eccleston. Whereas the Ninth Doctor preferred to be “coward rather than killer”, the Tenth Doctor has proved himself willing to be ruthlessly brutal in eliminating threats. In particular, when Rose is threatened he becomes almost pyschopathic in his determination not only to save her, but also to eliminate the threat entirely.
He also has an ongoing trope about his love of humanity, in partiuclar our inventiveness, ability to adapt and wild variety. He has had at least three big speeches on the inestimable potential and brilliant abilities of ordinary humans (once before the big bad aliens revealed in The Christmas Invasion, once before the Cyber Controller, and once before- well, I don’t think I’ll spoil that last one, but it is certainly the most terrible threat the Doctor has ever faced, bar none, and IMO the best episode of Doctor Who). Contrast this to Eccleston being frequently scornful of “stupid apes” and you add up to quite a different Doctor. He also does a mean Scots accent.
He does indeed return, in School Reunion, the 4th episode of the 2nd series (if you count The Christmas Invasion). Turns out he was left on earth as a gift for Sarah Jane. He is very important in that episode.
Well, not yet (either) but in “New Earth”, an otherwise forgettable episode, a dying Face of Boe reveals he has a great secret to reveal to the Doctor. Smart money on this has it being the survival of at least one Time Lord, hopefully the Master!
Yes on both accounts. This series is one of the most popular shows on the BBC, constantly dragging in by far the largest audience share for its timeslot, so I would predict new series for some time to come.