Be very careful with that first ride after derailleur adjustments. It's usually no big deal if the front derailleur drops the chain (except on climbs while using foot retention -- especially strapped into toe clips), or if the rear derailleur drops the chain off the smallest cog and wedges between the freewheel/cassette and frame. But if the chain drops into the wheel from the largest cog, that can get ugly. At low speed on flat terrain away from traffic, do several shifts from the middle to largest cog before venturing out into the wild.
I always test ride my bikes on the indoor Cycleops trainer after any derailleur adjustments, before taking a ride outdoors. I've found the trainer will reveal problems that don't appear when the bike is on a workstand with no pressure or resistance on the rear wheel.
Regarding chains, I've had better results with Suntour freewheels and friction shifting using plain plate narrow chains. The fancy chamfered/beveled plates that work so wonderfully with indexed shifting seem to want to ghost shift and rattle more on some friction shifting setups and Suntour freewheels.
I have a couple of Suntour 7-speed freewheels -- 13-24 and 13-26 -- and they generally run quieter and shift more smoothly using plain plate narrow chains, either an old school Shimano narrow or the older style KMC chains for 7 speeds. Unfortunately KMC has completely revised its nomenclature, renaming some chains (the Z-72 is now called something else), and either renaming or eliminating the old plain narrow Z-33 and similar chains.
When I use SunRace freewheels -- 13-25 and 13-28 -- I'll switch to KMC Z-72 chains. Bought a few new when they were still available under that name. They run quietly and shift smoothly with both Suntour and Shimano index shifting systems.