Hi Jim B,
welcome at Stridaforum 
The white one is a Strida 5 version (I’d not purchase an older version nr than this.)
The black Strida must be a model 3; these had the freewheel inside of the front beltwheel and there is no chance to replace a (most likely) worn rear beltwheel because that part was incorporated into the hub (one part and not separable). Some parts of this version are not any longer available - like wheels.
Things to look for?
Yes, many 
Rear hinge (Achilles heel), locking mechanism, ball bearings of the bottom tube next to locking latch, condition of the steering ball/plastic cap, check steering bars wiggle, freewheel condition, brake pads,…you may find some more typical issues of used bikes at this forum 
Genuine?
I’d say most likely are both genuine - but to be 100% sure I’d need more detailed images; for example showing the lettering on the belt, close ups of the brake calipers/rotors and the welding seams, plus pics of the bottom bracket region (The more, the better).
Cheers,
Chris