For your specific questions -
I have been in and out of chastity for the last year or so, currently back in with 22-24hrs/day, releasing only for shaving or sports. I have not had any issues getting erect when not in the cage nor have I noticed any changes in size/girth/rigidity. There are some hardcore chastity wearers at chastitymansion who have noticed shrinkage over many years (which has been their goal), however others have commented that the penis does recover even after years of constant wear.
It would depend on the clothing. General consensous is that a cage is not practical for bike riding. This is probably because it pushes your testicles forward and out. They can not retract like they normally would when cycling so you may end of sitting on them. As for running, this seems to be possible but you’d want to make sure your cage is snuggly supported (and not too heavy).
I go to a gym 3 times a week (free weights mostly) and play badminton, I don’t wear my cage during these times.
For only 4-6 hours at a time, you really wouldn’t need to worry. It sounds like your GF is only interested in using the device on you during playtime, a tease and denial sort of scenario rather than full on orgasm / penis control.
For the most part, my cage is symbolic and that it affirms my position that my penis is for Mrs Sens enjoyment only. Wearing it or not during things like sports is irrelevant in that its not a time where I would pleasure myself anyway, and why risk an injury?
To really get the “benefit” of chastity, for me I have to push myself through the mental barrier of being horny but without allowed a release, I can then surf the wave of being aroused, completely enamoured with my wife where your feelings of lust go back to the early days of courtship. For me, that takes about 4-6 days after orgasm to get there. My longest lockup is about 3 weeks.