OK to give a fake phone number on a short app or long app?

You're missing a lot.

 

It's hard to say exactly.. I have a few business odds and ends to tie up first. Was hoping in April/May, but it may take longer.
I noticed what you're saying about the recruiters. Is it because they don't want to incriminate themselves by putting their promises in writing?
There are certain things that I would like to know about certain companies now, so that I can come up with a game plan upon returning to the USA. For example, my wife needs a cdl and it's good to hear from companies which schools they'd be willing to hire out of. I got a short list of schools from one company, but then they followed up with a message saying, if there is somewhere else you are thinking about going for cdl school, run it by me and I'll let you know if it'll work. So, I sent two options we're considering and of course they haven't responded. They must have hired a private investigator to go visit the schools and see if they're up to snuff.

I do have unlimited minutes for phone calls to the USA, but my number is foreign and longer than US phone numbers. Most forms won't allow it. And some companies seem to refuse to answer when they see a foreign number. Some companies have even gone so far to have their IT department set up a firewall so that you can't access their website without a vpn routing your connection through a US location first.
I'm also not sure if it looks bad. Sometimes companies won't consider employees for stupid reasons like that. I've been living outside the US since the start of the plandemic, so I'll have to account for that time on my resume with "entrepreneur" or something more specific. I know a lot of companies would be worried that I'm more commited to my business that I have going on here vs. staying for the long haul with them, just like some companies look at you sideways when you tell them you're not interested in going to your residence for hometime. Some companies don't care, but others seem to want candidates to fit some cookie cutter mould. Bottom line, I don't really care about them knowing I live abroad with the phone number, because it will be reflected on my work history anyway. The main thing is that the forms require a phone number.

So, I asked my mom if I could use her number, and she agreed, but then when we learned how these recruiters basically blow up your phone, I feel bad for her. Most of my doubts for the companies are just quick questions and answers. If they actually have their little cheat sheet it would take them less time to copy and paste the answers than it would to pick up the phone and go through all the formalities of a phone call.

My first recourse is to just email them and briefly introduce myself, but as others have noted, they usually don't want to engage with email.

So back to the original question: What should I put for phone number?