Is it safe to run and charge a deep cycle battery in an enclosed space like a bedroom?
Not "safe" without extra care. Death unlikely but likely possible if you "just do it". 

Can be made "safe"

Discharge will not release 'fumes' in general use.

Risk from "gassing" (Gargoyle knows) can be kept small but not zero when charging with a panel of that relative size. ie 12v say &amp; 30 Watt = 2.5A. 24V = 1.25 A. + say 3% and 1.5% ~ of bat 1 hour rate.

Small but non zero chance of Hydrogen and sulphuric acid fumes if battery not designed to vent unless 100% sealed with full recombination control. Ventilation would be an extremely good idea. 

More if required ...

Long long long ago I charged a truck battery at a moderately high rate in a small closed room where I was sleeping :-). Took many weeks for my throat and nasal linings to "recover".



Data point: Just came across this [here](http://www.vonwentzel.net/Battery/00.Glossary/
) while looking for something else: 


Gassing: This is a very dangerous condition that can occur if batteries are charged too fast. One of the byproducts of Gassing are Oxygen and Hydrogen. As the battery heats up, the gassing rate increases as well and it becomes increasingly likely that the Hydrogen around it will explode.
The danger posed by high Hydrogen concentrations is one of the reasons that the American Boat and Yachting Council (ABYC) requires that batteries be installed in separate, well-ventilated areas.
" ... As the battery heats up, the gassing rate increases as well and it becomes increasingly likely that the Hydrogen around it will explode. The danger posed by high Hydrogen concentrations is one of the reasons that the American Boat and Yachting Council (ABYC) requires that batteries be installed in separate, well-ventilated areas." | From here | And ABYC site here


American boat and Yachting council website here
  http://www.abycinc.org/

No open access data available on AYBC site - you have to be a member to read about how to be safe.