If I had a cheap 35' boat that I wanted to leave somewhere for about two months for free how hard or risky would that be to do?
Pretty hard to do safely and very risky to do.
If I had solar panels and a reliable anchor light and a really good anchor could I just leave it out in the ocean somewhere a little distance off shore?
Yes, but it is a really, really, really stupid idea to do so....
Could I just find an anchorage and leave it there for an extended period?
Sure, but it isn't a good idea. If your boat breaks free, you're liable for any damage it causes, since you'd probably be considered negligent at the very least. Also, if your boat sinks, you could be found liable for any environmental damage it causes and the cost of cleaning up such damage.
The USCG limits the fees for environmental remediation to $800,000
, but that is a pretty big chunk of change.
What about moorings? I really know nothing about them. Do you usually have to rent one? Are there restrictions on the length of time if I found a free one? Can I make my own mooring?
Moorings are a far better idea than anchoring. Moorings are designed for long-term unattended boats. Most places charge for the use of moorings and if you found a free one, you would be seriously abusing the privilege of using it by tying it up for a long period of time.
Yes, you can make your own mooring, but you usually need to get permits and such for them. Any mooring you can make for less than the cost of renting one for three months is likely to not be safe or able to pass inspection.
I've heard various cities have time limits on anchoring. Would this be a problem?
Yes, it could be. Also, they could declare an unattended anchored boat abandoned or a hazard to navigation and remove it. If they do so, you'd be responsible for any associated fines or storage fees, and those are easily going to be more than renting a mooring would be in cost.
Is a $10,000 boat likely to get stolen? I don't want to pay $200 a month for a slip or dry storage. If the probability of the boat getting stolen is less than 24% per year than it is cheaper to risk having it stolen than paying for storage!
It depends... just remember, even if the boat is only $10,000, the parts are far more valuable. Someone could strip the winches, mast, boom, anchor, etc., off the boat and replacing them would cost NEW MARINE HARDWARE prices for a 35' boat. That isn't going to be cheap. I'd also point out that most insurance companies won't pay out if you're leaving the boat anchored out unattended for long periods of time rather than on a mooring or in a slip, since that qualifies in most cases as negligent behavior on your part.
I'd have to agree with some of the other posters and ask you to seriously consider if you can truly afford to own a boat.