Yes, and he should keep his boarding passes and any other documents evidencing his time in the Philippines. Hopefully, the proxy marriage certificate will satisfy the Philippine Consulate so they issue him a visa to enter the Philippines, AND, that he is fully aware of and complies with or is willing to comply with the local LGU protocols where you live. Currently there is no international tourism in the Philippines. Spouses of Filipinos can get visas, but still must follow the protocols in place at the place and time they'll be visiting. Many cities and provincial/municipal ports of entry are requiring RT-PCR tests and letters of acceptance. Would be nice if a lot of that goes away before May, but hasn't yet.
I don't see why you can't. Once you visit each other (thereby "consummating" the marriage) you'll be able to file.
If the thought hasn't already occurred to you, please make sure when he visits that you take pictures of the two of you together, with notable landmarks or signs in the background if possible.
