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I apologize, but I'm not exactly clear on what information you're looking for. If you mean that you want to develop a healthy root system on a lemon tree, following good cultural practices involving correct watering, light, soil, and fertilizer will result in a healthy tree, and that means healthy roots. This article has more information: https://www.gardeningknowhow.com/edible/fruits/lemons/how-to-grow-a-lemon-tree.htm

Pruning lemon trees isn’t generally required, but there are several good reasons to do so. Pruning helps improve airflow, reduces disease, strengthens limbs and makes it easier to harvest fruits. The best time to prune is early spring or late summer or anytime prior to blooming. Prune every year or two, which will help keep them from becoming too large. If you have frost damage, wait until the trees have shown new growth. Prior to pruning trees, pick off any remaining fruit. Prune all dead, damaged, weak or diseased branches to the base. Weak stems cannot withstand the weight of heavy fruit production.

Lemon and lime trees are quite similar, including their care. Pruning lemon trees isn't any different than pruning limes. The following article should be of some help: https://www.gardeningknowhow.com/edible/fruits/lime/tips-for-pruning-lime-tree.htm

Here are several articles to help you track down the source of your lemon problems: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-leaf-drop.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/growing-lemon-trees-containers.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-not-fruiting.htm

They will continue to ripen, but you will be sacrificing some flavor if you pick them early. A light frost should not cause harm to the fruit, but if you expecting a heavy frost, it may cause some damage to both the fruit and the tree.

They should not. Roots of most trees, lemons among them, are pretty lazy. They will grow where there is soil and don't like to force into things. Tree roots become a problem when the tree is lacking space or water. The roots then become aggressive and can get into pipes (looking for water) or push out walls (looking for space). As long as the side of the tree that is away from the wall is free from obstruction, it has plenty of space and will not push on the walls.

You can thin them if you want to, if the tree's not too big. However, the tree itself will usually drop many flowers, and even small fruit, before it arrives at a good number that it can develop to maturity. This article has more information: https://www.gardeningknowhow.com/edible/fruits/lemons/growing-meyer-lemons.htm