As a forester, I tend to take them at the ankles rather than climb.
consider where the bulk of your weight is and any lean the tree may have, weight higher up has more effect as it is further away, basic lever principle kind of thing. If you have the space to work with the dominant direction, go with it – if not, you need someone more experienced.
Tree 1 method, does not sound like a good idea. Avoid working at height if you don’t need to and taking roots out before the trunk would make things far too unpredictable. Roots will go a long way and make a big hole if you do manage to remove the stump, would consider a stump grinder or leaving a big enough stump to pull with a mini digger, but not too small, you’ll need the power – not looking so cheap, could just trim stump right to floor?
Tree 2, again I wouldn’t want to climb if not needed, this sounds like a straight tree and a straight fell. Guessing weight of branches is on your side so will come into your garden as long as the tip isn’t angled out over your neighbours.
Take this all as a vague guess, couldn’t be accurate unless I stood in your garden. Tree surgeons are for tight spaces where you don’t want things hit and for surgery works.
As Kuco says, this job sounds doable with a decent pruning saw such as a silky and some effort. pruning saws are very efficient and would be safer to climb with, or you could even get one on a pole to avoid climbing. Not a bad idea actually, about £50 will get you a pro grade one. These are shrubs in the scale of things and a Silky is an awesome trail saw.
