Hi, don't know how much experience you have growing anything, or bonsai, but in general hollies don't like to dry out, which means watering when the top of the soil is dry, but if the rest is still sodden from the previous time, you might need to repot into soil with more grit in it, to drain faster, or you'll end up with root rot (those 'mall-sai' are usually in cheap soil that ends up all compacted so the roots drown, and is concrete when dry). You can feed it Miracid every 10 days at half strength (starting in late March) because hollies love acid. Right now and for a season or 2, you just want to concentrate on keeping it alive and well, and the bonsai part (styling, etc.) will come later, and give you time to read up on things and/or find a local club to join. Good luck.

You might have gotten more response if you had copied the photos and all here to this forum. It's also been less than 24 hours and most folks have other obligations that may slow responses. Typically holly wood is quite white from everything I've seen. There aren't photos of the branch before removal, so it is difficult to diagnose what the problem was that caused the leaking sap. The dark wet spot looks to be a former injury that is healing over (see how the edges are rounded and look to be growing toward the center of the hole?) but trapping water. I am also not sure if the photos of the cut branch are of what is still on the tree or the part trimmed off. If it is still on the tree, it looks like the remaining stub is a bit long. Read up on proper pruning so that you are down near the branch collar but not cutting into the branch collar since that will create the best environment for healing. I can't grow holly here since we are so cold so don't have direct experience with this plant. I would be patient. Either it will heal or it will continue to decline, and time will answer that. There is not anything you can do to make healing happen faster. If you are concerned and there is space to plant a small replacement nearby, you could do that. If you add photos of the cut part that is left on the plant and the plant as a whole, not just the bark part of the branches, you may get some additional input on how healthy the remaining plant is.

It's a Burfordii, probably Burfordii nana. Burford hollies are a direct mutation of Ilex cornuta, Dan. The botanical name is Ilex cornuta 'Burfordii nana'. So they are a Chinese holly, though a mutation, a cultivar. To my eye, a Burford holly looks absolutely nothing like a yaupon, Ilex vomitoria. Leaf size, overall visual texture, stem color, nothing. You can't be all that familiar with the yaupons. (smile)

That really is a difficult picture, being largely in shadow and blurry. But I can just about see that it is alternate so it isn't any kind of privet. Not having berries would not eliminate holly. Being dioecious, only female plants would have berries and even then they could be stripped by wildlife.