As mentioned, I'd consider that wobble normal for an older freewheel block. If you have a lot of patience you could strip it and play around with shims and improve it but it will make no functional difference so long as the bearing spins freely.
As for the skipping chain, it does not look worn enough in that video to slip. My guess is that you have a stiff link(s) in the chain. Inspect each link to see that they all move freely and flex any that seem stiff from side to side to help free them out. Also check that the rear mech is free moving with good spring action and capable of taking up the slack in the chain.
As for fitting wider gearing, you say French? Is it a French made bike? If so, the threads on the hub are probably the French metric standard thread pitch so good luck getting one of those. Wide range 4 speed blocks of any kind are pretty rare. Your rear mech may not cope with bigger sprockets. The spacing is probably narrower than what you'd require for 5 or 6 speeds.
If you do need lower gears, it's probably easier to fit a smaller chainset. Or my personal choice to shove a Sturmey Archer AW3 in it and have wided gear range and easier maintenance.