Corporations can not directly give money to candidates which includes sending money to a 3rd party at the direction of the candidate. It's a much more serious violation that simply a failure to report. Usually reporting violations are dealt with civilly, though I couldn't confirm that there are no criminal penalties with a brief google search.

Those sound like very tenuous grounds. Most peoples' lawyers bill them for ancillary expenses, from printing/copying to first-class flights and other purchases in the line of work. It's not inconceivable that hush money would be in that category. Anything on a legal bill tends to go on most peoples' books as a "legal expense." In New York, there's no requirement to give clients itemized legal bills, although the client can request them. High-end lawyers tend to avoid itemizing their bills unless you ask (implying some sort of fee dispute).

At the same time, is this really the kind of thing you want a prosecutor to go after a former president and likely future presidential candidate from the opposing party for?

>Ironically, if he'd just paid it directly from campaign funds and given it an obfuscated name ("PR Services"), he'd be in much less legal trouble now.

No, he would have probably instead be indicted for using campaign funds to pay a mistress under the theory that such a payment is not a valid campaign expense. The rules are so unclear that you could prosecute him either way.