Well, let's tackle some of your list... there's a lot to cover and I don't think it's going to be a one-brief-post reply.
Let's talk about recording. Well, that's an entire thing. Any two-track recorder should work if you're just looking to record your stereo mix as-is. But if you're looking to track individual parts and mix later, it'll be a mash up of the number of parts versus the number of inputs you have in your recorder. If you have more parts than inputs, you might have to sync your modular gear with your recorder and make a few passes to get every isolated part.
Speaking of recorders, if recording to Ableton is out, then you could try the Akai MPC Live series with CV outputs. Maybe it's too close to a DAW for your liking. Else you can go with a straight multitrack recorder from Zoom, Tascam, Sound Devices. I have my eye on the Cymatic Audio uTrack24 recorder. The price seems fantastic (under $600US).
If you must synchronize, that's usually done via MIDI, so you'd need something like a Pam's New Workout and the MIDI break-out expander for it and a recorder that can sync.