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It is likely that your tree is grafted, as most citrus are. This means that the rootstock of the tree is different from the scion (top). Sometimes, rootstock will try and outcompete the unnatural top by sending out its own shoots. These should, really, be clipped off, or at least kept down to one or two branches, as they will steal nutrients from the kumquat, while feeding its own branches more.

This is the most likely, and most common, explanation for this phenomenon.

This sounds like it could be Black Citrus Spot.
This link will help you.

http://www.crec.ifas.ufl.edu/extension/black_spot/citrus_black_spot.shtml

Yellow leaves can be caused by many things including lack of nitrogen, insufficient light, water-logged soil (plant roots need oxygen to thrive), dry soil, or iron deficiency. ... If new leaves are yellow with green veins, the problem may be a lack of iron.
https://www.gardeningknowhow.com/edible/fruits/kumquat/growing-kumquat-trees.htm

I don't think the fertilizer sticks are adequate, they don't provide a full range of nutrients. Switch to a more complete organic fertilizer product like this:
https://www.downtoearthfertilizer.com/products/blended_fertilizer/citrus-mix-6-3-3/
or this:
https://www.domyown.com/dr-earth-organic-all-purpose-fertilizer-p-9485.html
Work the fertilizer lightly into the soil surface and then mulch with an organic compost material.
Deep water after applying fertilizer, and be sure to water adequately, deeply - about once a week, not daily or several times a week, citrus can get root rot if overwatered. Water the entire root zone under the foliar canopy, not just next to the tree trunk.

The white spots on the leaves are hard to distinguish, but may be an insect pest.

For one plant you don't need a lot of insecticide, so you can buy a "ready to use" spray bottle like this:

https://www.domyown.com/monterey-houseplant-insect-spray-p-11260.html

or organic program alternative:
https://www.domyown.com/bonide-insecticidal-soap-rtu-p-1667.html

Depending on that tree was propagated, there may not be a graft point. It still looks an awful lot like the original tree.

Still... It is a very low branch, and will be more of a nuisance than it is worth later on. If it were me, I would cut it off and only let new growth grow on the branches above the fork. This will keep the shape of the tree, and preserve its single trunk later on.

This article will give you more information on the care of these: https://www.gardeningknowhow.com/edible/fruits/kumquat/growing-kumquat-trees.htm