Chain skip options on vintage 'frankenbike'
First, I'd trouble shoot why the chain is skipping. Does it only happen under heavy load? Does it always happen in the same place (maybe a stiff link)? Does the derailleur do something funny? 

If the BCD and bolt pattern matches, stick whatever chainrings you want in there (subject to front derailleur constraints on chainring size + tooth gap). As for if your chainrings are worn or not, the picture isn't there, but there are plenty of other questions (see sidebar or search) on this SE to check if your chainrings or cogs are worn beyond. 

While using a 9 speed chain on a 7 speed cassette should work, it will likely be a bit sloppier than using a 7 speed or 8 speed chain (they are narrower than the chains that the 7 speed cogs were designed for). They're also pricier. So, I'd also go back to a 7/8 speed chain in the future. 

The change from Suntour freewheel to Shimano freewheel shouldn't really have caused any problems other than maybe some minor adjustments (unless you're running a microdrive drivetrain).