There’s no real definitive way to train reactions, but what you can do is train your muscle memory to respond to certain attacks. Reactions is something you develop over time, with real match experience, and is not going to be an overnight process. Set your training dummy to walk back and forth, throwing out pokes, and jumping it at random times with different attacks. Try to pretend as if you’re playing a real match, so play footsies with the dummy, and once you see the jump-in, do an anti-air. The anti-air doesn’t necessarily have to be a DP, it can also be a normal move such as a cr.hp (Ryu) or a cr.mk (Abel), anything that can prevent your opponent from jumping on you for free. As far as hand positions go, they should be in whatever position you feel comfortable with holding the stick. While it is easier doing inputs starting from a neutral position, you will not get this luxury very often in a match. Practice doing your anti-airs from a standing or crouch blocking position, since that’s where you’ll be most of the time.
The same training dummy, with the random jump in attacks and walkign back and forth motions, can be used to train continuous cross-ups. A good training dummy would be Ken, since he has an excellent cross-up (mk). Try to get a feel for what the spacing looks like when Ken jumps in. Defensive SRKs can be practiced in training mode by yourself. If you can do it ten times in a row, in training mode, then you should start a basic exercise with your training dummy. Set the dummy to trip you, and then jump in with an attack on wake-up. Try to do a reversal DP, FADC, backdash. After that you can set the dummy to do 3 crouching light attacks, walk up throw. This helps teach you how to do it from a defensive crouch position when you’re being put in an opponent’s attack string. All the problems you have can be worked out in the training room.
Try to focus on one or two characters. If you invest some major time in a select group of characters, you’ll quickly see improvement. If you play several different characters, it’s difficult to get a feel for how specific matchups should be played out. Familiarity with your character, and matchup knowledge, take time to develop and start gaining the wins, and it’s difficult to do that if you’re constantly switching characters.
Once you can do these things in training mode, consistently and without error, you’re ready to try it out in real matches. You’re never going to learn something completely until you’re able to pull it off in a match. It’s easy to practice motions or setups in training mode, where there’s zero pressure and you’re in complete control of the situation. However in a real match, you never know what the opponent is going to do, but the very least you can do is prepare for their options and setups by working it out in practice, so that you’re ready.
Think of it like doing drills for a sport. It’s repetitive and tedious, but it help develop your fundamental skills.