Q.Should I be pruning my 10 month old (citrus) lime tree?
So my nearly one year old lime tree has just been growing more and more branches, which keep getting longer and longer – so much so that some of its really long branches are beginning to arc and over the past three months, I’ve noticed that 2 branches have split (with one of those two completely breaking off after a while). I tried loosely tying the trunk to a stake in the middle since I wanted it to grow straight, but I don’t think it did much. Should I be taping/tying these branches back together or should I prune the lime tree at this stage so it doesn’t grow all wacky and out of control? Or is there something else I should be doing to this lime tree? Below are some pictures of the lime tree as a whole, its arcs, and its branch splits.

Certified GKH Gardening Expert
This looks quite normal for a vigorous tree. You may prune at any time, since it also looks quite young. It isn't completely necessary, but can help with the shape.
This article will help: https://www.gardeningknowhow.com/edible/fruits/lime/tips-for-pruning-lime-tree.htm