Hi, this is Assaf.
For a few linear maps(pl), and most of the control point maps, especially with few people (so you don't get noticed), the pyro's backburner is actually a good choice, since you can use the aternate route and surprise your enemies from behind. A great example of this is dustbowl stage three, with the tunnel that only spies use. The idea is that on defence, you go there, bump into a few spies and kill them from the front, then wait at teh entrance. After a few people go by, and you realise that these are the only ones that have spawned (Tip: wait for the slower classes too), come out and flame only when near enough. This way, you put the attacking team in a kind of a trap since they can go forwards and get killed by your teammates/sentry or they could face you and expose themselves to your teammates pushing.
For ctf and busy servers don't bother with the backburner, since rockets will be coming at you all the time, and all alternate routes are used.
About the flare-gun: It's shit. It deals very low damage and takes extreme accuracy, like the demoman pipe bombs, except that they explode and can hit multiple people, and they can roll. If you are using the flaregun for long range, chances are you are firing over the front line, orelse you would have used the backburner. Think: Behind the front line,w aht are there? Dispensers and medics. Hence, flaregun = useless. The shotgun is also a much better choice since it fires linearly, and the shots are not affected by gravity. Because of this, your chances of hitting are much higher. Not to mention that it does way more damage than the flare gun. The flare gun is only good for maps with no medikits like warpath. But still, you CAN find medikits lying around, and the fires do eventually go out. For Arena, where there are no med packs, flaregun is your choice, unless you are facing another pyro. If you are confronted by another pyro, you need the shotgun, badly.
The axetinguisher: very very awsome. It works very well witht he pyro's compression blast, since enemies can get pinned against walls like that. Not much to say about this weapon, since it's so awsome. The axetinguisher has no downsides imo, since no one uses the regular axe anyways.
I don't expect anyone to read this, but still...
~Safibn