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It would be very common for the Thai Lime tree to be grafted onto a root stock, and I do believe that is what your seeing. A growth from the root stock. There is no way to really know what was used as the root stock unless the grower you purchased from would have a record of the root stock that was used.
From my research Trifoliate orangne is a common root stock, from which the fruit is not edible, but it certainly could be something else.
I would remove the branch, as it can zap the energy from your tree.

I have included a link to Thai Lime Tree care.

https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm

Citrus trees should be watered with clean rainwater whenever possible. Tap water is often treated with chlorine which can be harmful to lime trees. If your tap water is treated with chlorine, you will want to invest in a filter to remove the chlorine from the tap water. If a filter is not a possibility, you can leave the water you use on your citrus tree to stand uncovered for 24-48 hours before you water your tree and the chlorine should evaporate to harmless levels. If your tap water is hard (alkaline) you will want to add a tablespoon of vinegar to 1 gallon of water to lower the pH.

Placing gravel or small stones in the bottom of the container will help improve drainage. i would use a potting mix made for citrus or you can make your own with a mix of equal parts soil, sand and peat moss.

For more information on the care of kaffir lime trees, please visit the following link:
https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm

What you are seeing after pruning is very normal. Pruning encourages branching and a more bushy plant. That long wayward branch can actually be pruned again! I would recommend continued modest pruning over leaf picking at this point. I would reserve leaf picking for when the tree is more mature. If you decide to pick leaves, do only a few because your tree is not very leafy at this point.

For more information on the care of kaffir lime trees, please visit the following link:
https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm

Without a photo, it's hard to say, but it sounds like maybe your kaffir lime might have bacterial leaf scorch. Here is an article on leaf scorch for your review:

https://www.gardeningknowhow.com/ornamental/trees/tgen/bacterial-leaf-scorch-disease.htm

Happy gardening!

You can trim away the dead material and wait and see.
Here is a link to refresh you on the care.

https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm

This is Sooty Mold.

Here is a link that will help you.
https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm