I've played against a tink in tournament and talked to Jash at MLG. He said basically for bombs, shielding stops bombs dead in their track. Pshield them so your at a frame advantage. Pshield his arrows, and boomerang and punish accordingly.
Do not focus on catching them, focus more on spacing. Zair you also pshield this and punish if he spams it.
The air dodge bomb trick is easy for you as a player to over come your just not adapting to his tricks in game. He does it once or twice, you can bait him to do it punish. Or you can wait for him to get out of his air dodge and punish with a grab or an attack. Pummeling can make him drop the bomb. If the bomb does go off he gets punished and you both take damage. I've run into situations with tink before where i get him in a grab and his boomerang breaks it, and we just chip damage each other. If you run into a situation where you both are back into a neutral state re-collect your thoughts and plan your next attack.
Your next paragraph on when you say your the one taking damage once you get inside. I think a sonic main can help you here, and help you with learning to deal with bair. Alot of learning how to play a fighting game is knowing what to do at any one point of a match given certain factors. So for my 2 cents on helping u here i would have to say at low 0%-50% go for sonics easy to combo into moves at low %. For higher % focus more on punishing the opponents DI, getting good reads, punishing
any of his mistakes. Without getting hit very much yourself at high %.
I think Sonics better get off me move is fair. But someone else here could tell you different. I dont play sonic. But i know how to deal with tinks his projectiles very well.
Heres an example of me dealing with Tink on my DDD.
http://www.youtube.com/watch?v=C7YOZBtHEl0#t=2m20s
http://www.youtube.com/watch?v=EYNsP_6yTNw#t=4m32s
Alot of the mu is really about spacing his projectiles and yourself.