Agitation may help on commercial size systems (maybe), but it could only cause negatives in homebrew IMO. It will add oxygen, and the possible for contamination.
In my home brewery wild yeast means anything that wasn't purposely put into my beer but is there now. If I pitch a vial of Brett, it wouldn't be wild IMO.
Going to secondary will not change the amount of sugar you need to add. Unless you let the beer sit over 6 months, you will have enough yeast to carb your beers. However on big beers or beers that have sat a long time, I will pitch some fresh yeast from a starter or vial. It doesn't take much...
Rouse the yeast and warm it up. If that doesn't restart fermentation, then create a starter with a well attenuating yeast and pitch it at high krausen. Good luck.
Looks pretty normal for biab that I have done on that scale. Next time you may want to give it some time in the kettle after cooling to let some of the trub settle before going into the carboy. Give it 4 weeks to ferment and there should be a solid layer of trub and yeast at the bottom. Let...
I used leapinlizardlabels.com for some homebrew labels for my wedding last month. The labels came out great and were exactly what I wanted. As long as you know the size and have a good image, they are really flexible in sizes and types of paper. Shipping was reasonable and I got them really fast.