This is a guitar from 1950's or 60's that I recently got refurbished, but now I still have the problem of it going relatively quickly out of tune. It has a movable bridge, and I think this is the reason. Is there any downside if I screw it in place? Why would it be movable in the first place if it's only making trouble?
That is a floating bridge, typical on archtop guitars. Under no circumstances should you "screw it in place." It is meant to be adjustable so that the intonation of the instrument can be set. It should not be possible to easily move it when the instrument is strung; if it is, something is wrong.
The bottom surface should conform to the shape of the top, giving good surface contact for transmission of vibrational energy from the strings, and this should provide plenty of friction to keep the bridge in place. If the bridge is out of alignment laterally, or if there is a dramatic mismatch between the top and bridge surfaces, it seems possible that the bridge may move when bumped.
I doubt that your tuning issues have anything to do with the bridge. If the intonation is off, you need to reposition the bridge, and may want a luthier to do this (though it is certainly possible to do yourself). If you do remove the bridge, you may want to check to see how well the bridge mates with the top; if the fit isn't tight, you can adjust by placing a piece of fine sandpaper to conform to the top and carefully sanding the bottom of the bridge in place.
When you change strings on this guitar, unless you are removing the bridge, change only one string at a time to avoid moving the bridge. And if the you do remove the bridge, mark the location (with some tape, e.g.) first, so that you can reposition it more easily. You may consider placing some tape (maybe some light-tack blue painters tape, no duct tape, please) near the bridge in a few locations to help diagnose any actual movement while playing.
It could be that you need to change the way you install strings. To minimize string wrap at the posts, I always tie my strings on; you can find ample information about how to do this on the internet. Tuning strings to the desired pitch from below seems to work better than from above for equalizing the tension on both sides of the nut. Make sure that the tuning machines aren't slipping. Failing that, you may want a luthier to look at the nut to ensure that it is cut correctly.
Difficult to tell from the angle of the photo, but I'd say the bridge needs moving towards the tailpiece. To check intonation, look elsewhere on this site for instructions - it's been well covered. Suffice to say the 12th fret (and 19th fret) harmonic needs to be the same pitch as the 12th fret note (and 19th fret note) fretted.
The bridge on these guitars is moveable to allow differing intonations depending on the string guage. Not strictly true - as it's easier to make an archtop and put the bridge wherever later, as once it's fixed, there's no going back.
EDIT: the strings have recently been changed. New strings always take some settling in. It may well be that they weren't 'stretched' enough and that's the cause of untuning. Give each a good pull, they need to sit properly at the nut and bridge, and if there are several turns around the tuners' posts, they will settle in there too. Without checking the intonation, the thing will never be o.k. fo rtuning. Don't take anybody's word, get it right yourself - it's not difficult, even without a tuner, which I tend not to use.
Do not fix the bridge mechanically, or with glue. The strings will hold it in place if it's played properly, assuming the profile of the feet match that of the archtop at the correct position.