"The Law" absolutely runs on "when did this piece of paper arrive at this location". I don't know about DCMA specifically, but yes, most cases where one actor claims another actor is doing something wrong require the reporter to follow specific steps or it's not valid.
This is nonsense. Google cannot simply provide a fake or broken button and get away with it. If Google is required to provide a mechanism for reporting DMCA violations in a timely manner, and they do not, they're going to found liable no matter what kind of excuse they provide.
The reporting buttons in Play make absolutely no claim that it's for DMCA violations. Because it's not for DMCA violations.
DMCA claims go elsewhere, e.g. mailing them a physical letter or going through their separate reporting system for legal violations: https://support.google.com/legal -> https://support.google.com/legal/contact/lr_dmca?product=goo... . And, honestly, I'd still trust physically mailing them over that form - mailed letters start legal time limits for actions in many (many!) cases, online methods rarely have court-tested equivalent protections.