No problem.
: Sometimes the jump to the first line is better, but in this shape I think the solid connection is better. The reason is easiest explained by looking at how the actual game turned out. After
, black can take B8 in sente, which is quite useful for him (this isn't sente if you block at C2). By playing C2 instead of C1, you may give black the first line sente hane-and-connect, but that's not as big as letting black block off the left side in sente.
: The problem with that sequence is that it leaves your Q4 group weak and exposed. I'm not sure what the exact best move is for black here, but let's say he plays something like Q8, which seems pretty good. Now you're white group is getting surrounded and will have to struggle around a bit to live (of course it can live, but you may have to play some small moves or moves that help black to do it). Meanwhile, all of the black stones around the J3 stone are very strong, meaning J3 only serves the purpose of making territory, without attacking black in any way. Invading for territory is okay, but it's better when you can simultaneously threaten your opponent while invading.
So, in my proposed variation (Q2 at Q3), white has a lot more eyespace in the corner, and the black group has less eyespace and more weaknesses to exploit with an invasion around J3. Also, because your group in the corner is stronger, it has the automatic effect of making the black R8 stone weaker.
: In this situation, I would've directly helped white's group on the left, probably with a move like J10 or K10. The group is rather large to sacrifice, and letting black get more moves to attack it doesn't feel all that good to me. I think it's better to firmly secure this group first before trying to profit from the black stone on the right.
I hope this helped. If you have any other questions just ask
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15 would you cut as white or play as in the game. And if white didn't cut would you run the stone out, or take the corner? And of course, would you just do something else entirely.