Two more factors to consider:
- Soil pH. Given that you have so much moss, your soil is probably acidic.
- Species selection. Different lawn species have different levels of shade tolerance. Don't plant "full sun" species in your mostly shaded lawn.
A strategy for having a healthy lawn:
- Start with a soil test. (MSU extension; $12 as of Jan 2011. See that page for sampling instructions.)
- When the test results come back they will tell you what you need in terms of fertilizer and lime.
- Get the mowing under control. If your neighbors can't get their lawns to grow well either, you should be able to get together and convince your HOA to apply some sanity to their mowing. (Possibly by getting the HOA to switch to a landscaping service that actually knows what they're doing, or getting permission to opt out of the service and do your own mowing.)
- Improve your soil. This is probably the hardest part, because if you're working with heavy clay, you may need to amend it heavily to get grass to grow well.
- Is bringing in topsoil an option? If your yard isn't too large, and you have the budget for it, several inches of good topsoil could do wonders for your lawn. (Obviously, if you decide to go this route, don't bother with a soil test on your existing clay!)
- Otherwise you can aerate, amend with compost, and fertilize regularly as needed. It will take time but you can gradually improve your soil this way.
- Seed mix. When you seed your lawn, look for a shade-tolerant mix.
A completely different strategy: Don't grow a lawn.
I know someone who has a situation similar to what you describe -- shady, moss grows well, poor soil (they're too sandy instead of clay). They have a decent size yard, but a tiny lawn. They have perennial beds instead: small trees, shrubs, flowers, ground cover. Mulch with shredded bark. It can look beautiful (better than lawn) and it can be lower maintenance too. It doesn't need to be done all at once -- you can convert your lawn to gardens over time, a new area each year.