Charging deep cycle

I've been using a Ctek like that for years and it seems to work fine.

Current battery is about 5 years old and still holding charge well. The charger runs through it's full cycle.

Short answer is YES , long answer brings into question size of charger and power draw of the Engle , bottom line is that charging the battery will take longer while drawing power from it at the same time. Other than that no drama.

No drama. As long as the charger puts in more power than the engel takes out.

as others have already said- YES
i charge my second battery in the back of my rodeo whilst the fridge is still running off it. i have a 105ah allrounder and a 10 amp 3 stage charger. It does take a bit longer but that to me is no hassle. I have it set to float at 13.8v, when the compressor on the fridge kicks in it goes back to absorbtion mode then back to float soon after compressor stops.
I also use to leave the aux batt connected to the main battery via the isolator, i noticed it would charge both batteries up at the same time. for some reason when i feed power back through the cable to the isolator from the aux batt end (via charger) it opens the circuit up. i would have thought once the main batt voltage drops to 12.8 (shut off voltage) that the 2 batts would be completey isolated but apparently only in one direction (main -> aux). I didnt realise this and cooked my main battery a few months ago because the charger was set to calcium-type batt and was outputting up to 16v (which is correct for the allrounder deep-cycle), but the wet cell up front didnt like it much. Just one thing to take into consideration i guess. Perhaps check to see if your isolator (if you have one) allows power to flow back to main battery.

hope that helps

cheers