It's a very good question because it reflects how reality sometimes differs from what you're taught in the books and in training.
As with most things, it depends. Let's take your two questions in order:
1. Take the active or wait? Depends on the kind of aircraft approaching and how far out they are. You have to consider his approach speed, actual distance (since base-to-final turn is a maneuver, not a fixed position in space) and what happens between you both if he has to go around (will he fly up your tailpipe?). If it's a Cub and he's doing a spamcan circuit, you've got all kinds of time to safely take the runway and depart, especially if he's still on base. If it's a Pitts and he's just turned final, you've got all of about 10 seconds before he's over your head, so I'd wait it out. Plus, if he has to go around, he's going to blow right past you on his climbout. If someone is on base leg in a standard circuit, yes I'd go just about every time. In most cases, they've still got about 2 miles to cover and you'll have no conflict at all. If someone is on final, I almost always wait it out, for courtesy as well as safety. For the sake of a few seconds, it's just not worth it.
In all cases, though, no screwing around on the active with final checks. Do all that behind the hold line. If you're not ready to firewall the throttle, you've got no business being on the runway yet.
2. On downwind. If I've got the performance (Pitts) or the distance (number 1 is on a long downwind), I'll just cut inside him with a short downwind, which means a short final. If it's a spamcan doing the typical bomber circuit, I can usually be on the ground and ordering a cup of tea before they're crossing the hedge. I'll make the call on the radio that I'm turning base inside the Warrior, so the pilot knows what I'm doing. In both the Pitts and a Cub this is no problem. With the Pitts I'd have a huge speed advantage. In the Cub, I've got great slow-speed advantage and can drop like a brick with a full-on slip so I can make a very short, steep final, and then slow to exit, sometimes before the first turn-off from the runway, which means I'm clear of the active in plenty of time. Key here is to judge your speed and distance differential and get off the runway as soon as possible. If you're going to do this, you can't force the other pilot to change his approach at all or you've screwed up. This is probably a maneuver best left to experienced pilots and at your home airfield where you know the conditions and typical pattern behaviour.
I prefer both of those to extending downwind (wait until Number 1 turns final and passes you, then turn base). That just makes the problem worse for anyone behind you and on it goes. In the Pitts, it hates long, flat finals. Can't see jack over the nose and you have to drag it in on the prop. In the Cub, it takes forever to cover a long final like that. Bores the hell out of people.
Short answer - yes your instructor is being overly cautious. Good for teaching purposes. Probably worth a discussion about his logic to learn why he's doing it.