Double:
Why lat pulldown + deadlift instead of wide grip chins + deadlift: first I think it'd be too much of a strain to do both efficiently in one workout (and the deadlift gives you more overall benefit), second it's hard to gauge progress with bodyweight exercises because your bodyweight changes constantly for some reason or another. They're an excellent strength exercise but hard to measure progress with. Lat pulldown with a heavy enough weight will widen your back, fear not.
It all depends on the kind of weight you're pushing and how much weight you'd have strapped to your belt do your chin-ups with.. I prefer rack chins to ordinary chin-ups, they fry the lats more and you're not really hefting much of your own weight there so it's easier to progress. Chin ups aren't nearly the beginning when it comes to back width.
Silverwex:
Your chest doesn't feel sore? That's not necessarily a sign that you didn't work it out enough, once you've set your baseline weights and are able to crush these every workout then soreness is a mere triviality. Don't worry, when the weights go up you'll get sore! Don't sweat it, add some weight next time and continue.
As for your wrists, it means you're not positioned under the bar properly so as to be able to hold it with no wrist involvement, somehow you're putting pressure on your wrists here. Are you doing these with a free barbell or a smith? Are your elbows strictly tucked in? I dunno how holding the bar improperly as Lifeforce suggests can be an issue, I've always ever held the bar one way.. maybe I misunderstood.
2-3 sets or one rest-pause set is fine for shoulders, they're not different to any other minor muscle group (like biceps and triceps).
Lat pulldown is wide overhand grip, your arms are just hooks, pull with the lats.
Chest alternatives: you have flat db press, incline db press, decline db press, flat barbell press (if you HAVE to), incline barbell press and decline barbell press.. basically you want any two of these exercises at any one time, so it's unlikely that you'll have stagnated on all six at a time.
Can you make your flat bench decline by propping up one end on a box or block or something?
If you've started with barbell, stick with barbell. Sorry to Lifeforce but I don't agree with changing exercises just because they don't make you sore. If you can progress with them then the absolute worst thing you can do is change them for something else. When and only when you get stuck on one exercise, and I mean you can't progress any further in weight, do you change - soreness is neither here nor there.
Phew. All in one breath, that.