Regarding your 3rd question about the hard 1st to 2nd gear upshift - just feel lucky that your car starts out in first gear.
The biggest complaint about these W124's is that they have poor acceleration. All pre '92 W124's start out in second gear. 1992 is the first year of first gear start, and it was a big improvement. The rear end gears are also better spaced apart, and you get a higher 4th gear for freeway cruising.
I think all transmissions shift with authority from 1st to 2nd gear, so yours could be fine. I do know that this can be adjusted. There is an adjustment in the bowden cable (which I've never done), as well as a vacuum modulator on the transmission that can be adjusted to make the shift firmer or softer. The adjustment for this has been posted before, so you can try a search. I'm sure M.B. DOC and Stevebfl have commented on this.
Also, just know that when cold, your car has an upshift delay valve that is there for smog reduction purposes. It delays the shift from 2nd to 3rd gear, and holds the car in 2nd gear to a higher rpm, until the catalytic converter warms up.
2001 E430, Bourdeaux Red, Oyster interior.
1973 280SE 4.5, 170,000 miles. 568 Signal Red, Black MB Tex. "The Red Baron".