dray gave some good answers. I'll chime here in with my opinions.
1) I sold out my arena tonight on the first match of the season. I don't want to spend money building the arena bigger quite yet, so I was thinking about raising the cost of tickets. how much do you think I could raise them by? I stomped on my opponent's in the match by 40 points so I'm hoping that the fans will be happy, plus next match I'm playing a team that is really weak and hasn't been logged in for a few weeks.
I would definitely raise the prices. Perhaps not to the max, though, but if you're selling out, then the sweet spot is certainly above your current prices. I doubt anyone knows exactly how far. I suppose the easy answer is to say that if you've raised prices and still sell out, then you didn't raise them enough. :) The other easy answer is to look at other teams that have started maybe a month before you and that have had early success - your crowds will be similar to theirs, but a month behind, so you can probably improve on their profits.
2) What do you tend to tell your coach as to the depth chart? I've been keeping mine around 'coach picks from depth chart'. What are the advantages/disadvantages to having strict/liberal coach orders?
I tend to go with the default option - let coach decide - mostly because I'm used to it. The only time I really move away from that is if I'm really focusing on training, like if a trainee needs to get 48 minutes of playing time at a position, in which case I'll put the player in as starter, backup, reserve, and have the coach strictly follow the depth chart.
3) How important is it for teams to 'take it easy'? What impacts the rate of enthusiasm declining? In hattrick the coach's leadership dictates how quickly team spirit will drop over the course of the week.
Taking it easy is pretty important. It's similar to PIC in Hattrick in that it gives your team enthusiasm a boost and playing with higher enthusiasm helps your team perform better. I forget exactly what the enthusiasm impacts - I think it helps rebounding and defense. Someone who knows more can correct that. As far as the enthusiasm levels falling naturally, I would assume that the trainer's skill level impacts that, but I don't know that for certain.
After winning my first match and (hopefully) getting some good trainees and a better trainer, I'm feeling like I understand the game a bit better and am enjoying it. I'll just need to remember to keep setting orders since there are so many more matches each week compared to HT!
Congrats on the first win. :)
There's a lot to get used to here, but it eventually starts to make sense. And yeah, setting orders three times a week is something to think about. (And if you get supporter, you can play in a private league, which gets you an additional game each week. I actually like this feature a lot, as it allows me to see how my team performs in various situations without any negative effect.)
Good luck!