mileskb

First... as others have said,  use the Helix as your interface.  If for no other reason it cuts down on the number of conversions your signal goes through.
 
Regarding compression and EQ, while the general rule is to record as dry as possible, that's also considering the "old school" technique where you are mic'ing an amp in the room to record.  In this scenario, while some engineers do it, you really shouldn't put any EQ or compression on that mic that is doing the recording. However, there certainly may be compression and EQ on the front of the signal chain to achieve the sound in the room, and you'd want to leave those in place.
 
With the Helix, things are a little different.  If you are putting the Compression at the END of your signal chain this would be like putting compression AFTER the mic that is recording... you don't really want to do that.  It's not part of your sound.  If on the other hand your compression in up front, and part of your sound, than by all means leave it in place.
 
Same for EQ.  I would turn off Global EQ, and I would not use any EQ AFTER the cab for recording.  Again, you may have some EQ up front in the signal chain to get your sound.   Use of compression and EQ up front some people refer to as "sustain" and we wouldn't want to kill that.   
 
An easy way to think about all this is...  Everything BEFORE the CAB in the Helix chain is equivalent to "in the room" or what's coming going to be coming out the speaker.   Everything starting with, and  AFTER the CAB in the signal chain is more importantly AFTER the Microphone that's pointed at the Cabinet in the signal chain.  

Every day I practice at home on computer with backing tracks (for now I used Line6 UX2 audio card).
OK

Every week I go to the rehearsal music studio with my band. I use an existing amp + cab. It is not always the same (Fender, Blackstar, ...) so I should rely too much on learning the settings of a specific one.
If possible just tap into the power amp.  Skip the pre-amp.  Sometimes accomplished by plugging into the FX loop return, or AUX in, or Line In.  OR....
I would assume you are using a PA at rehearsal to?  Just send the outs (or out if mono) to the PA and send another line out to the AMP section of whatever is on stage to use as a monitor or just have the PA route the signal from the PA to your stage monitor.

Every month I go to a concert where I usually plug directly into the PA.
have them route some of your guitar back to your stage monitor or bring along something and provide a send to it.

 

If you really want simple...  get a powered monitor like a QSC 10 or 12 or one of Line6's units.  Frankly any decent powered monitor is likely to sound better for you than any of the options you are currently using.  This is what I would do if I was you..

 

Every day I practice at home on computer with backing tracks (for now I used Line6 UX2 audio card).
Use monitor.  I run my "backing tracks" into my Helix... so if I want to listen through headphones I can, or if I want to hear it from my monitor, I can by just turning it on.

Every week I go to the rehearsal music studio with my band. I use an existing amp + cab. It is not always the same (Fender, Blackstar, ...) so I should rely too much on learning the settings of a specific one.
Use monitor as stage monitor, send signal to PA

Every month I go to a concert where I usually plug directly into the PA.
Use monitor as stage monitor, send signal to PA

 

This way... same setup, every scenario and you always hear yourself the way you need to.