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If the branch sprouts off from the base of the tree or below the graft ring, you could be seeing the rootstock trying to regrow. Typically these kinds of branches should be removed from the tree because the rootstock fruit is inferior to the scion (which would be the Meyer's lemon for you).

This article may help:
https://www.gardeningknowhow.com/ornamental/trees/tgen/can-grafted-trees-revert-to-their-root-stock.htm

This page will help with IDing the type of pest attacking your fruit. It is specifically for identifying pests that cause damage to the skin or insides of citrus fruit. I think you will find it very helpful:
http://www.ipm.ucdavis.edu/PMG/C107/m107bpfruitinvert.html#DISTORTED

Usually blooming occurs in late winter to early spring.

White oil should not adversely affect the plant; however, it sounds like you may have an issue with mites or scale. The use of neem oil is very effective for treating these pests and is perfectly safe for plants, people, and even beneficial insects. Here is an article with more information: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

I don't think you need to worry with thinning out the blossoms just yet. Wait to see just how many fruits the tree will set first. Then you can simply thin out the new fruit as needed if there seems to be an over abundance, leaving just enough for your personal use.

This can happen either due to poor pollination or due to stress to the plant.

If the flowers are poorly pollinated, then this can be corrected by hand pollinating the tree. Just take a small paintbrush and swirl it around inside a few of the flowers.

It could also be caused by stress to the plant. The plant needs to maintain itself before it can maintain fruit. So if it is not getting enough water or nutrients, it will abort its fruit to save itself. When you see flowers start to form, increase the amount of water and give it a slow release fertilizer. That will help it hold on to more of its fruit.