I'd hold off on the oil until well after the bloom. Commercially, oil is used June through August, but seldom outside that range.

Yes, this cool weather is good for moving a tree. Keep the roots moist throughout the process, and really flood the soil where you plant it. I'd prune back the top so it has relatively few leaves to support. I'd not worry much about the flowers; it will naturally drop them if it needs to.

Feeding -- yes, now is the ideal time for your first application (old growers always say after the full moon in February). Then again in 4-6 weeks, and a third application in the fall, after the summer rains are about over. Avoid fertilizing with anything leachable during the rainy season -- it will just wash away.

If you move your lemon, yes, let it grow a new flush of leaves before you start fertilizing it.

Because you have sooty mold, start looking for the insects that generate the honeydew which is required by the sooty mold.

The potential culprits are aphids, scale, whiteflies, and mealybugs. Scale are the best at hiding out; they blend in well.