If you train near your limit, you should have restdays. To train everyday can stress out your nervous system. At the beginning u will make progress, but on the longterm, your muscles and body need the rest. As push ups are a strength exercise u should rest enough. But if u train skills like Handstand or Balance stuff that u can train everyday without problems

On the negative your lat is holding a bit, but u cant compared it with pull ups. Pull Ups are the mother of backexercises. If u can always do pull ups for back

On Dips your chest and tricep takes over, because they are stronger then your tricep and they take over the full load maybe. You can try different things. 1. pre exhaust your chest so that your tricep takes over 2. Concentrate on your dip technique 3. Look if looking your arms get you a better feeling or its is better for you to let the tension on your tricep(dont fully lock out) 4. Try to hit your tricep with another exercise 5. That you dont feel yout tricep fully doesnt mean its not working. To the wrist strain concentrate to hold your wrist in a straight line while dipping. Dont bent it to much, warm your wrist up and stretch it. Use wraps and if the pain doesnt go away switch dips for a time and do other exercises. Later u can come back and try again to dip

Soreness is not the only sign for building muscles. Soreness is not a must have to build muscles. More important is the muscle activation, consistency and that you make progress and overload the muscle. Soreness will come with new exercises/progression/new weights but after your muscles are used to exercise, you wont be sore everytime

I could do Pull Ups from begin on. At the beginning I could only do 2-4 Pull Ups. Then I did years of other sports and now started 6month ago with calisthenics again. Now I can do 20 Pull Ups in a row, hope more to come🙏🏻