Monroe is normally the OEM for the 1BA standard dampers. Sachs is normally the OEM for the sports dampers (1BE for A model, 1BD for S model). It appears Audi used both for the B9.0 1BE rear dampers, though. Back to just Sachs for B9.5.
As for the fronts, it appears that's what the 53 to 49 change in Jul '16 was about, when the 1BE damper changed from Monroe and the 1BA strut fork size to Sachs and the 1BD strut fork size.
original 1BE strut 8W0 413 031 AK by Monroe:
https://www.ebay.com/itm/276369895108
later 1BE strut 8W0 413 031 BA by Sachs:
https://www.ebay.com/itm/334813790034
Keep in mind that these represent the middle axle load code tier. Without you saying what your actual axle PR codes are, we can't talk the specifically relevant front dampers for your 1BE suspension PR code vehicle.
Bilstein B4 are twin-tube designs like the OE dampers (though seems Sachs OE dampers for B9 went monotube?). Bilstein B6 and B8 are monotube designs. I have B6 on my B8 1BA A4. There's nothing too rough about it. I assume you're thinking of B8 dampers, which are shorter and intended for aftermarket lowering springs. Personally, if I had a B9 without adaptive, I'd be trying the B6 DampTronic.
On the Sachs struts, 318141 vs 318142, you need to know your front axle PR code to know which to get. ...141 for if you had damper rev AT (codes 0JA-0JE), ...142 if you had damper rev AK or AK (codes 0JF and up).
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