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Yes you can. You might lose some lemons, but you shouldn't lose the tree. It depends on your skill at handling the soil-root-ball during the transplant. If you can slide it out of the existing pot and just loosen some of the circling roots without major root cutting or root loss, and if you don't allow the root ball to break apart as you place it in the new pot and backfill, then you will cause only minor transplant stress. If you can't slide it out, then break the old pot to get it out, don't break the root ball. Get an extra pair of hands to help in holding it together if you need to.

https://www.gardeningknowhow.com/edible/fruits/lemons/repotting-lemon-trees.htm

It probably won't grow true to type from seed. So you may not get a lemon like you are expecting. However, if you'd like it to branch, you can try cutting the top off about 6 inches to encourage branching. If new sprouts form at the top you can select a new leader and cut away the others.

https://www.gardeningknowhow.com/edible/fruits/lemons/propagating-lemon-seeds.htm

They will, more than likely abort. There isn't much that you can do about it. Leaving them won't hurt, as they will likely just fall off anyway.

This article will offer more information on the care of lemon trees: https://www.gardeningknowhow.com/edible/fruits/lemons/how-to-grow-a-lemon-tree.htm

Your photos did not come through, unfortunately. It sounds like both are suffering from root rot. Most types of root rot fungus are not treatable once you notice the above symptoms. This is caused, mostly, by soil that does not have a chance to dry out between waterings.

You can make an attempt at correction. A fungicide will be in order, along with using a combination of DOLOMITIC LIME and WETTABLE SULFUR. Follow all product instructions

This article will help you with fungicide use: https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

In the case of complete tree removal being necessary, you will want to treat the area, and NOT plant any more fruit trees in that spot for a few years.

These articles will help: https://www.gardeningknowhow.com/edible/fruits/peach/peach-phytophthora-root-rot.htm

https://www.gardeningknowhow.com/edible/fruits/citrus/phytophthora-root-rot-in-citrus.htm

From the photos it appears that the tree was pruned very heavily, creating a vigorous growth response. And/or the rampant growth could have been accentuated by high nitrogen fertilization, chemical or manure.

You can bring the tree back into balanced growth with more moderate pruning. Thin and shape back the long leaders by 25% of the length and no more than that at one time.
If you are not sure about how to do this, consider hiring a professional for the pruning.

Don't fertilize any more this year, but in spring you can take up a spring and fall fertilization program with a slow release, complete organic citrus fertilizer.

https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-pruning.htm

https://www.gardeningknowhow.com/edible/fruits/lemons/learn-about-fertilizer-for-a-lemon-tree.htm

As long as they are close to ripe, then they will continue off the tree. This can take around 9 months, or so.

This article will give you more information: https://www.gardeningknowhow.com/edible/fruits/lemons/harvesting-lemons.htm