Best way to run a long ethernet cable along walls semi-permanently without it looking terrible or getting damaged?

The answer to 1 is self-adhesive trunking, not clips, and the reason is load spreading.

A clip holds the cable at one point, so the entire weight of the run plus every accidental tug lands on a 15mm square of adhesive. They fail all at once on a warm day exactly as you describe, because the adhesive softens and the load has nowhere to go. Trunking is a channel with a lid, stuck along its whole length: the same adhesive over a hundred times the area, and the cable sits in it rather than hanging from it.

It also solves the vacuum cleaner problem, which clips do not — a cable in a closed channel with a rounded profile gets bumped rather than caught.

For rented plaster, the removal trick is heat and patience: a hairdryer along the length softens the adhesive enough that it peels rather than tears. Doing that on cold two-year-old adhesive is how people take the paint with it.

Doorways: go over the frame, not under the door, and it is not close.

Under a door the cable gets stood on every time somebody walks through, and a door bottom is a shear point — it does not crush the cable so much as work it back and forth until the conductors inside fatigue. That failure is horrible to diagnose because it is intermittent long before it is dead.

Over the frame the cable is out of everyone's way and the architrave usually gives you a natural shadow line to hide it in. Run up the side of the frame, across the top, down the other side. It sounds like more work and it is about ten extra minutes.

If you genuinely cannot go over, there is flat under-carpet cable designed for exactly this, but it is a different product and you want it by design rather than by squashing a round one.

On question 3, ethernet is more fussy than it looks and two things matter:

Bend radius. Do not fold it round a corner. The rule of thumb is a bend no tighter than about four times the cable diameter, so roughly 2-3cm radius for ordinary patch cable. A sharp crease changes the spacing of the twisted pairs and that is what the whole design depends on. You will not see a dead link, you will see a link that negotiates at a lower speed or drops under load, which you will blame on your router for six months.

Tension. Do not pull it taut between two fixings. Leave it slightly slack. A taut cable transmits every knock to the connector, and the little plastic latch on an RJ45 is the weakest thing in the run.

Other than that it is robust. It does not mind running alongside furniture, and for a 20m domestic run it does not care about mains cable nearby nearly as much as the internet thinks — twisted pairs reject that interference by design. Crossing at right angles rather than running parallel for metres is enough.