There are 3 legal requirements to enter as a tourist:
1) proof of funds (20,000 baht or equivalent in another major currency, IN CASH ONLY)
2) proof of onward travel (a ticket out of Thailand within the number of days you'll be stamped in for, 60 days in most cases)
3) proof of lodging (at least a couple days hotel booking)
Most people won't be asked for any of those things, and they generally only ask if they're looking for a reason to deny you entry or have some suspicion about you.
20,000 if you're from a visa exempt country, 10,000 if you're from a visa on arrival country. So 20,000 for most people. Only asked if they want a reason to deny you entry.
I told you what a re-entry permit it is. It keeps the stamp you have from being cancelled when you leave Thailand. If you leave Thailand without a re-entry permit your visa or extension will be cancelled. A re-entry permit does not give you any extra time in Thailand. It is not an extension. It just keeps your existing stamp alive.
If your extension stamp in your passport ends July 15th and you leave Thailand with a re-entry permit and return on June 1st, you'll be stamped back in on your existing stamp until July 15th. If you leave without a re-entry permit then you have nothing anymore as everything is cancelled. If you leave with a re-entry permit but don't come back by the time your stamp runs out then you also have nothing because you didn't return before your extension expired to apply for a new extension.
It is NOT a multiple entry visa. It is a re-entry permit, either single re-entry or multiple re-entry.
I'm confused about why you're asking about the DTV and remote work and also asking about a work permit. These 2 things are mutually exclusive.
Your contract doesn't have to be from your home country. It just can't be from a Thai company because the DTV does not allow working in Thailand, working for a Thai company, or working for Thai clients. You cannot get a work permit with DTV.