Thanks for the tips so far, guys. Some good additions for me there. It seems for my situation, my main needs are minor cuts/scrapes/punctures that a farm dog just runs into that I can treat at home, and then since Steve sometimes goes into the boonies with me, the potential for major wounds. Anything in between, and I can get him to his vet or the urgent care.
While I have the quikclot gauze, Dogmush brings up a good point with his canine trauma pack. While my own backcountry IFAKs have stuff like zfold gauze and Israeli bandages, I wasn't thinking about applying those to dog emergency first aid, and they certainly can be. On the topical antibiotics, my vet told me that the triple antibiotic I use is safe, though that discussion also led to her giving me the healing oil I mentioned. I just threw some chlorhexidine in the Amazon basket.
Interestingly, I found there are quite a few dog first aid training classes available, including by the Red Cross. Many of these appear to be online, but I'm going to take those classes. I'd still like to find a hands on one, as that's the best practice, as I learned when I took Stop the Bleed, where you actually had to "stop the bleeding". Plus stuff like having an expert available to guide me through simpler stuff, like apply vet tape, where I'm always worried about applying it too tightly while Steve slips it off if I do it too loosely.
On the vet glue, I'm talking about dermabond, vetbond, or any of the "medical super glues". On that, the clippers are a great suggestion. I've only had one bad cut on Steve where I wanted to close the wound before I got him to the vet, and stuff like steri-strips absolutely DOES NOT work with fur.
EDIT: Oh, here's an example link for online training:
https://www.redcross.org/take-a-class/first-aid/cat-dog-first-aid