Yes, pruning makes trees stronger because it fosters proper growth, weight distribution, and balance of the crown. By increasing the structural integrity of the tree through pruning, it’s less likely to experience disease, pests, broken limbs, and storm damage. Pruning the right branches and limbs is key to improving a tree’s strength and balance.

It is possible to cut down a tree on your property yourself, but it is generally not recommended unless specific conditions are met. You should only consider this project if the tree is small (e.g., under 4-5 inches in diameter and short enough that you don't need a ladder to reach the branches), and you are experienced in handling a chainsaw. The tree must also be located far from any power lines, fences, homes, or other structures. Even for small trees, the work can be dangerous, and it's always advisable to have a spotter to ensure the tree falls correctly and no branches fall on you.

For any tree that is large, requires a ladder to access, or is near electrical lines or structures, you should always hire a professional. Attempting to remove a large tree yourself is extremely dangerous, poses a risk of serious injury, and could result in significant damage to your property.

Before any tree removal, it's also important to check your local and state regulations to ensure the tree is not historic or otherwise protected by your city.

Unfortunately, palm tree stumps are one of the most difficult types to remove due to how their roots grow. Unlike other trees, palm tree roots grow horizontally and vertically. Palm tree roots grow as deep as three feet into the ground, although soil and growing conditions could mean they go as far as 50 inches into the earth. 

If you want to be able to repurpose the grass in the area where you removed a stump later, you’ll have to dig quite deep—and perhaps wide—to trim away all the roots.

If it’s your property, then you can plant a tree near power lines, but it’s not the best idea. Any vegetation that grows within 10 feet or so of power lines will fall under the responsibility of the utility company, and they may trim or remove it as necessary. If you do move forward with your plan, try to choose a smaller ornamental tree, plant it as far away from the power lines as you can, and keep it trimmed to avoid future problems.