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It sounds like it may have a fungus. Treat the plant with a fungicide and this should help.

In all probability, that extra branch is being produced by the root stock onto which your Meyer lemon was grafted. The usual advice is to prune off any branches sent up from the root stock, but I'm wondering, did you try the fruit from the rogue branch? If you liked it, you could let it grow instead of pruning it away. However, generally you would want to prune it off, to let the tree concentrate its energy into producing the fruit you want. This article has more info on grafted fruit. https://www.gardeningknowhow.com/ornamental/trees/tgen/can-grafted-trees-revert-to-their-root-stock.htm

Cold weather can have some strange effects on plants, and it can take 2 or 3 years for everything to get back to normal. You can certainly do some pruning on your tree. This article has some general information: https://www.gardeningknowhow.com/edible/fruits/lemons/growing-meyer-lemons.htm

Re: potting soil recipes, you can use any bagged potting soil, but I prefer not to use anything with "moisture retentive" additives. Then I like to add a quantity of perlite, 1/2 - 1/4 the volume of potting soil. Please understand that there is not one "best;" every gardener finds their own favorite. The most important thing always is to make sure the soil drains well. Here is an article for you on growing lemon trees: https://www.gardeningknowhow.com/edible/fruits/lemons/how-to-grow-a-lemon-tree.htm
container gardening: https://www.gardeningknowhow.com/edible/vegetables/vgen/quick-guide-to-container-vegetable-gardening.htm
and some basic soil information: https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetable-garden-soil.htm
Re: fixing used soil, I'd recommend adding a variety of organic improvements, plus perlite for drainage. You can look for specific fertilization guides for each kind of vegetable, but generally speaking, a good well-drained soil, plenty of light, sufficient water without drowning roots, and a balanced fertilizer used with a light hand (plants make their own food remember - fertilizer is a helper, and can easily be overdone) is the basic formula for success.

Meyer lemons often flower more than once in a season, so your flowers are perfectly normal. I would leave the flowers on the tree, because they smell so wonderful, but don't worry about pollination -- you want the tree to be putting its energy into growing roots and branches rather than more fruit. For the same reason, you might want to remove all but 2 or 3 or the little fruits. At this young age, the tree won't be able to support all those fruit to maturity. I'm sending you an article on growing Meyer lemons, and also some more information on fruiting. https://www.gardeningknowhow.com/edible/fruits/lemons/growing-meyer-lemons.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-fruit-drop.htm

Patience is probably what you need. It can take awhile. This article has some more information you might find useful: https://www.gardeningknowhow.com/edible/fruits/lemons/growing-meyer-lemons.htm