I use the Demoman and Pyro about equally.
1) Why do I use the Demoman: He's a defensive monster. People may hate sticky bombs, but only because they are effective. I enjoy putting some stickies around corners or on ceilings. I can take out a wave of enemies all at once. The pipe-bomb launcher allows me to take out sentries that are around corners or on a higher level. And when I feel like something different, I can switch into Demoknight mode.
Why do I use the Pyro? I'm versatile with the class. I always (and I mean always) go with this loadout: Degreaser, Flare Gun, and Axtinguisher. I use the hell out of my compression blast, which gives me an edge in many ways.
2) The class that gives me the most trouble:
As a Demoman, I'd say Soldiers give me the most trouble (though they aren't at nemesis level. They have to be a good Soldier). They can rocket jump and sneak up on me from behind. And once they are close, not much I can do to them won't also damage me (I could melee them, but thats hard to do while you're being rocketed around.
As a Pyro, only other Pyros give me problems consistently. This is especially true when they carry both or one of the shotgun and Backburner (the Backburner does more base damage and flares don't ignite Pyros, making the shotgun a better midrange choice). I usually take the cowards way out and blast the Pyro back and try to escape, as my loadout simply doesn't do enough damage against them.
3.) What strategies you use to attempt to beat your opponents with your favorite/most used character?
The compression blast. It's what separates good pyros from bad ones.
Defensively, I concentrate on guarding buildings and separating a medic from whoever is being healed (using the compression blast). My favorite thing to do: Find a Heavy+Medic and run up behind them, using my compression blast to forcing one of the two into the range of a sentry. I can completely nullify an ubercharge with the compression blast.
Offensively, I focus more with the Flare Gun. I try to find people while they are still far away and then set them on fire (and send them into panic mode). If they decide to flee, I'll continue shooting flares at them (and get mini-crits from hits). If they decide to fight, I'll still shoot flares until they get close, at which point I whip out the Axtinguisher and usually take them out in one chop. For Heavies (who don't usually go down in one chop), I alternate between the Degreaser and Axtinguisher once they get close to me. I blast them in the air, switch to the Axtinguisher and chop, and repeat the process until they are dead. Soldiers usually end up eating their own rockets. And Demoknights are useless against me.
4.) Any extra thoughts or opinions you might have.
Demoman and Pyros are definitely my main. The third most time is actually spent as an Engie, although I hate being an Engie (I don't like having to stay in an area guarding my buildings. It's boring). But I'll switch to the class if my team is doing a poor job defending and doesn't already have an Engineer. I hate being a Sniper for a similar reason. I know the Sniper can go out and be more of an attacker on the move, but I can never shake the feeling that I can fill that role way better with just about any other class.
I've spent the least amount of time in game as a Heavy. I hate the lack of mobility and the fact that the class is a target for everyone. I feel vulnerable when I play the class.
My up-and-coming class is the Scout. I use to never use him because I played on a laptop before, and the trackpad made aiming really hard with a fast-mover like the Scout. Now that I've been playing with a mouse for a while, I've gotten the hang of him and have been using him more.
