No, There's absolutely no limit on the time someone can take to issue an invoice.
10 days is a very short payment demand time though, particularly when it's taken him 6 months to actually issue it. Normal practice is more like 28 days.
It's perfectly feasible he could be an entirely legitimate business and still have cheques paid into his own name. It's not illegal. Tens of thousands of businesses around the country aren't VAT registered because they don't have a high enough turnover. Nothing suspicious in that either.
So no, all of the points in the original post are irrelevant (though as I said 10 days is a short demand and, unless you previously agreed to that, I probably wouldn't fall over myself to pay it within a month unless you want to).
On the other hand, the subsequent post contends the work was actually unsatisfactory. If that's the case you are out of my knowledge and into the hands of the lawyers. I certainly wouldn't look to pay for something that wasn't done right.