Agreed that it is a totally normal thing to feel like you're taking forever compared to people who've been doing this for a while. My biggest tip for time management, especially at your stage, is to start everything early and give yourself plenty of time to do a good thorough job. If it takes you half an hour to do a full assessment, give yourself the half hour, starting early enough that you're still starting your feed (if applicable) on time. This will keep you and the kiddos on schedule. As you get more efficient, it'll take you less time, and you can start a little later.
As far as your actual assessment, it sounds like you've got things pretty much in the right order. I like to go from least invasive to most invasive, so I'd start with counting resps (yes, using the hand method, or just watching the chest if it's visible enough) and doing a BP, then a temperature so they don't have much of a chance to drop after opening the incubator doors or unwrapping. I agree that the thermometer almost inevitably causes crying, but a pacifier or some containment holding should settle them down well enough for you to proceed with auscultating.