Yes the second sound is the most disturbing.....
sounds coming from the rear when turning (slow turn but sharp turn) can only come from the diff (the inner diff; the lsd itself) or tyres.
Analyzing the wormgears in the diff itself might be very difficult.
Going downhill and getting noises can come from a number of things. Final drive gears, propshaft, gearbox, propshaft support bearing etc.
But to be clear: the shop that installed the diff only installed the complete lollypop from the other car? They did not physically install the lsd?
What might be a good idea is getting the lsd out of the lollypop and put it into your own old casing, if you can't localize&define the sounds to the propshaft or some bushing or so...
That way you know the crown&pinion are ok (I assume your old diff has reasonable mileage and has not been tampered with), you know the crush sleeve from the pinion shaft hasn't been tampered with (very important!), the bearings are ok etc etc etc.
Because the biggest uncertainty is that some other bloke has been there before. That can be a good thing.....and it can be a very bad thing.....
Be sure that your shop has experience with the bmw 188 diff in particular. The bearings are a hassle to get off with lots of potential risk for damage and they cant be bought new (at least, not easily).
Fitting a complete diff (the complete lollypop/casing) is a whole diffenent ballgame than fitting an acutal lsd into a casing.
For the swap you need new oilseals on the outgoing axles, and the rear lid paper gasket is not sold anymore so you have to use some blue silicone sealant.
The shop has to have special tools for fitting the lsd.