The Scarlett would be a good choice for recording. Here are some issues you may encounter: all audio interfaces introduce delay between the time the sound hits the interface until its recorded and played back through the DAW. This is called latency or sometimes referred to as lag. Most mid-range and higher USB interfaces offer the ability to mix the direct sound (your synth) and audio coming from the computer so you can monitor without lag. As long as you're not trying to listen to what you're recording through plug-ins IN your DAW this is an acceptable solution. You can get interfaces with very little delay (practically zero) but the cost goes up substantially.
The other issue is that Eurorack synth level versus line level (the level of operation in mixers and like gear). Synth level is hotter than line level. It's very likely that you can just adjust the input volume on each Scarlett input to compensate. Do your homework if this will work with the Scarlett. Ask around.
If you record each part on its own track (you'll need two inputs for tracks in stereo), you will have them isolated. You can process isolated tracks separately and is the best way to mix/edit your music. If you record everything as one stereo file you are basically stuck with what you have. Any effects (EQ, compression, reverb) is applied to everything in the mix... which is something that you probably don't want.
If you're playing live without a DAW, you'll have to decide if you want/need EQ, compression etc. You'll also need some way to provide them. I don't see too much of this inside of most peoples' racks. But they'll often use an external mixer that has EQ built in. There are several types of EQs and a reason to use each of them if the material requires it. Without getting into a big philosophical, technical diatribe, I wouldn't worry about it in the beginning. If it sounds okay, it is okay. Having EQ, compression will make your live mix better. But if you don't know how to use them properly (for now) it's more trouble than it's worth. The exception would effects like reverb, delay, chorus, distortion, etc. They are very important in defining your synth sounds and shouldn't be overlooked.
Next chapter... SYNC. aka how to lock your Eurorack tempo to your DAW's tempo for overdubbing more parts into the DAW for later mixing/editing. Do a little research and report back. :)
There's a lot to learn from your starting point and no easy way to explain it without you doing a lot of hands-on by yourself. You may... and I say may... want to subscribe to an online course regarding mixing and DAWs since there's so much to explain. You'll also get a better understanding about signal flow between your modules and your DAW.