Carefully pruning small roots no thicker than three inches in diameter won’t harm the tree. However, cutting larger roots may impact the tree's stability and nutrient absorption. It's advisable to consult an arborist or tree care professional to evaluate the root's size, the tree's health, and the potential impact of cutting. Proper tree pruning techniques and timing can minimize damage, but larger root removal may necessitate expert guidance to avoid compromising the tree's overall health and stability.

Ideally, you’ll have a good enough relationship with your neighbors that you can kindly mention this task in regular conversation. However, every neighbor (and neighborhood) is different. Your best bet will always be to ask directly and nicely, offering to contribute to the tree trimming efforts if you’d like. However, if you are bothered by the tree and your neighbor is unwilling to cut it, you can involve the help of the homeowners’ association or local authorities, if applicable.

Make sure the tree roots soak around a day before you dig the tree up for the transplant. Keep the roots wet while the tree is above ground. After the transplant is finished, provide consistent water every day for the first month or so. You don’t want to overwater the tree, but you do want to keep the roots wet so they can thrive in their new environment.

It depends. Smaller, younger trees take longer to mature, but they're more resilient, their roots don't get as damaged during transplanting, and they recover faster. Plus, they're much more affordable to plant, and it's possible to DIY the planting. 

On the other hand, mature trees produce fruit faster and provide more shade and privacy immediately. Ideally, a small or medium tree is the better choice if you can wait a few years for it to mature. But if you need immediate maturity, a larger tree is the only option.