I have the same deal in my current bands, but luckily I play on distortion channel 80% of the set. If I do switch to clean I sometimes leave it on as I like how it gives a bit of dirt to the sound (I usually push the crunch button and turn the gain down to give it some breakup). So its not like I play on a squeaky clean clean sound, I like some dirt there. You could use the volume knob to roll off some gain as well, sometimes I do this if I'm not acting like a lunatic out in the crowd haha.
I haven't used a compressor on the dirty channel live so I can't really help on that, someone else will probably know and advise better than me.
As for the delay you can still run it in the chain, it'll just be in front of the amp that's all, so place it after your distortion and into the input of the amp. Some people actually prefer it in front, personally if the amp has an FX loop I use it via that. But if using, say a combo amp I don't have any other option.