Hi Buck and family, welcome. Your future posts won't need approval so post away! Just curious but what was the reason for the amputation?

Sorry your vet didn't give you good instructions. That sure doesn't help! I know this all seems so overwhelming. We created the Tripawds Start Here page to help ease you into finding the information you need, as well as Jerry's Required Reading List . Hop on over and check 'em out! Meanwhile I'll try to answer some of your concerns and questions:

 He is a bit swollen but seems more gravity collected fluid in his chest/arm pit area.

Very likely a harmless seroma . Click on the link for tips.

They did mention that alternating warm & cool/cold compresses could help with the bruising, swelling & weeping.  I justforgot to ask specfics how often, how long, how warm/cool/cold.  

See: 

Heat and Ice Therapy Tips for Tripawd Cats and Dogs

 things i forgot to ask (pill schedule for today & walking instructions [no backyard]

Your vet needs to tell you the pill schedule. Just curious but what medications did they give you? As for walking, he should only be on short leashed potty walks at least until stitches are out. Keep activity to a bare minimum, rest is medicinal.

if there’s a post or video with instructions on materials(I’m thinking of cutting up a terrycloth bath towel) preparing (is it just warm water (tap, distilled, filtered?) placing & time frames for compresses I would be very grateful.

See the massage link up top for tips on that. You can certainly use a human warm/cold compress, just make sure you have a towel between it and your dog's skin.

And then there’s the question of cover — should I put a shirt on him now or wait a few days when he’s feeling less sore?

Most dogs go commando unless it's super cold, there's kids in the house who might be traumatized by the incision, etc. 

Let us know if this helps. Feel free to call the Tripawds Helpline too, we have folks on live this evening but you can call and leave a message any time. Stay tuned for feedback from others!