Hi, I want to process the line level audio signal from my desktop synth through a eurorack filter and then to a line level desktop mixer. Question: Is an amplifier before and an attenuator after the eurorack filter necessary? Would it influence / improve the sound quality?

Yes, it's necessary. Your synth's line-out will either be at .775 or 1.4V maximum, and in practical terms lower than that. Audio signal voltages in modular synths, however, run anywhere from 5 to 10V peak-to-peak. The result without an input preamp will probably wind up not being all that useful, noisy, etc. Also, if you want the sound to do any control functions (ie: envelope following, gating, etc) you'll need an input module that allows for this by having a built-in envelope follower. This would allow you to do "auto-wah"-type processing by patching the envelope follower to the VCF cutoff.

As for something on the output side...if your mixer can handle the levels noted above, it should be OK. But at the same time, you're connecting a lot of different devices here, and every time you add a new one, your risk of ground loops and noise goes up. It's always best to send your output signal to the outside world via an output module that both attenuates the signal back down to proper line levels AND also isolates the mixer and modular from each other to prevent those noise issues.

One other note, also...if the input module's envelope follower also outputs a gate signal on the input reaching a certain level, you could then use that gate to trigger an envelope generator to impose an envelope on the VCF. For example, the EG could then control the cutoff, and the envelope follower output itself could then do something like contouring your VCF's resonance -- the louder the signal gets, the more the resonance increases, etc. That can be pretty useful, and another argument for using a proper input module.