Re:
You are reaching the parts where access to bike-specific tools is likely to significantly decrease your quantum of frustration.. I dunno if you've got 'em- chain-link extactor, crank extractor, BB spanners, cone spanners, spoke key..?
127mm should not limit your options too much- not around here in retroland, anyway! But, rather than a screw-on freewheel, that sure looks like a 'cassette' to me, which is outside my zone of familiarity, even with six cogs. With six cogs, I guess it is from a time when cassettes were still a novelty, possibly subject to yearly innovation, so there might be compatibility issues with trying to get seven cogs on there, or availability issues with the seven (obsolete?) cogs that will fit on there? I don't know. Someone does....