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A lima bean will not grow into a tree. It is a vining annual, so may get several feet tall but will never get woody like trees and will die after a few months.

If you planted a lima bean, then yes, there is a good chance that if it does not die before it is full grown, it will grow more lima beans.

Obviously, someone must have mislabeled them, as bush beans do not typically behave this way. Beans with runners are known as pole beans and do require some type of support. These articles will help: https://www.gardeningknowhow.com/edible/vegetables/beans/growing-bush-beans.htm
https://www.gardeningknowhow.com/edible/vegetables/beans/pole-bean-supports.htm

This is normal. You can make it a bit more bushy by nipping off a 1/4 - 1/2 inch of the end of these runners. This will force them to "bush" out rather than up.

From your description poor pollination is your problem. Try planting additional plants and flowers that attract pollinators to help with that issue. You might also want to research using Mason bees for your garden. They are great pollinators and do not sting. There are bee houses and bedding available both commercially and you can make your own.
You have been misinformed about using fertilizer. Plants need it to grow, you get too much vine from to much nitrogen in the soil, normally it's from over fertilizing. More is NOT better when you are apply this amendments. By adding additional organic matter I'm sure you are getting most of your fertility from that, but when in doubt, don't guess, soil test.

You can store fresh picked lima beans at about 32 degrees F. with 90% humidity for about two weeks. You can also freeze them or can them, depending on your preference.

This article may help: https://www.gardeningknowhow.com/edible/vegetables/beans/bean-blossoms-no-pods.htm