OP, if you are planning on cutting down trees, lowering your garden AND paving it, where will all the rain water that is currently soaked up by your garden go?
Paving a previously unpaved area is very environmentally unfriendly and adds massively to water run-off. That's why there are now strict rules about paving front gardens to create parking spaces. It was causing roads to flood, because the storm drains couldn't cope with the increased amount of water run-off.
You also need to bear in mind that trees take up a lot of water, whihc will have to go somewhere else if you chop that tree down. Someone near where my friend lived cut down some trees at the bottom of their garden and, after heavy rainfall, many of the back gardens of the houses in the street that ran along the back were flooded, some so badly that the water got into the houses. The insurance claim went well into six figures. Admittedly, they were 6 mature trees and the gardens behind were 6' or so lower than the garden where the trees were, so it was a much bigger job than you're proposing, but you need to be sure that you've taken full account of the impact of the work you're proposing.
And when you say the trees are in your garden, this is only the case if the whole of the trunk is on your side of the boundary. Are you sure that the post you're talking about was originally a fence post, and not up for a washing line or some other purpose? Plus you can't really judge where the boundary is from one post, as sometimes gardens are wider at one end than at the other, so the boundary line isn't perpendicular to the gardens.
This sounds like a significant piece of work and, given that your neighbour appears to be unhappy about it, I would at least get a proper survey done by a landscape architect before proceeding, check out exactly where your boundary lies and who is responsible for which boundary.
I think this sounds like a significant project and you don't seem to have thought through all the implications of what you're proposing. When my BIL terraced his sloping garden and laid a large terrace at the top, plus felled 2 trees because of disease, he had to put in some serious flood mitigation measures, including digging a soakaway.