it's helpful, if he's visa exempt, he doesn't need to use the visa website, whether the website is working or not. (The TDAC website is a different one).
I think you can be in Thailand while his visa is processing. He can't, tho, as he needs to remain under the jurisdiction of the Embassy, which is never Thailand.
The next 90-day report date is not a visa. You're legal until the original date, unless you get an extension, which also isn't the same as the 90-day report.
In your case, you can leave until August 4th, but if your stays goes beyond July 30th, then you need to file another 90-day report.