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If you feel the tree is not setting fruit due to pollination, it can be hand pollinated. Simply take a small paint brush and swirl it around in the blossoms and move from blossom to blossom.

If you would like fruit, then yes they will need to be pollinated. If the lemon tree is small, you may want to hold off until the lemon tree is larger. But if it is large enough that you feel it can support fruit, you can pollinate the flowers by using a small paintbrush to swirl inside the flowers and this will pollinate them.

As long as you live in a climate where it can survive outdoors, it can be planted next to a wall. Walls are great for citrus because they bounce back heat, which the plants love. You will need to prune it to keep its branches from rubbing against the wall or you can choose to espalier it (https://www.gardeningknowhow.com/garden-how-to/projects/espalier-instructions.htm). Lemon trees are not aggressive and will not harm your wall, but you just have to take steps to make sure the wall does not harm the tree.

This is referred to as vivipary or a viviparous mutant. It happens because Abscisic acid (which keeps the seeds from germinating) in the fruit either break down or are not produced in high enough quantities.

Lack of Abscisic acid can have a few causes. One would be unusual exposure to cold. Another is that the fruit produced or the plant itself has a mutation. It can also be caused by nutrient deficiencies, which would happen after neglect like you describe.

Since the tree has not done this before during its fruiting life, this is likely that this is a one time issue. Resume proper care and it will correct itself.

When growing fruit, citrus trees do need quite a bit of water. I would not cut back water for them when they are fruiting, especially when they are in containers. In the ground, they can access water from deeper depths, but in containers, you are their only water supply and it is just best to keep them well supplied for best growth. I would suggest regular water while flowering and more water while fruiting, especially early on when fruiting.

It sounds like the tree has a pest, maybe scale or aphids. It is hard to say as there are several pests that cause this kind of damage. We recommend treating these kinds pest with neem oil. It is safe for humans and animals but is very effective against most destructive plant pests.