If that's what you want to do, why not? To answer your questions, it depends why the 'companion' horse or pony is not currently ridden, some might have health problems that would exclude too much physical activity but most, even if a bit arthritic or stiff would benefit from some gentle physical exercise. Yes you can take a horse out for a walk in hand if you want, people often do it when starting youngsters or rehabbing after an injury, you should wear a hat, gloves and hi-vis out on the roads as you would when riding (and make sure horse is safe in traffic and not too spooky before setting out- you absolutely do not want horse pulling away from you or spinning and getting loose!) - you can also long-line in hand on the roads which would give you more control than just a lead-rope.
Re a share, I personally wouldn't part-loan/share my horses with someone who only wanted to do in hand work/agility/walks with them, but that's because mine are fatties who need to be worked pretty hard to keep their waistlines trim. You my well find someone with an older or injured horse who would welcome you doing this particularly if you are prepared to chip in with stable duties as well.
Do be prepared for some prejudice against horse agility, personally I think horses are a hobby and so long as it isn't prejudicial to the horse's welfare you can do whatever you like, but there are a certain kind of 'wafty', parelli-acolyte, advocates 'natural horsemanship', into lots of in hand/loose schooling person who gets right up the nose of the more traditional horsey type - to be fair there are a few of these on my yard and their horses are a bit bargy and rude, they are too scared to ride them, fitting the stereotype, but then again there are some traditionally trained/managed ones who fit the same bill so who knows 
!
I think if you did want one of your own you'd be well suited to look at a rescue - the larger national charities and lots of smaller ones too have plenty of small, easy to keep young-ish ponies - these are usually marketed as non-ridden/companion only but usually it's not because of any physical issue, it's just the rescue lacks the time and resources to back them properly, and/or they may have been (badly) started previously, and the rescue doesn't want any bother from people who imagine they can pick up a cheap/free ridden pony they can plonk their novices kids on no bother. If you choose carefully you could definitely find a pony who'd love doing agility and going for walks etc - although as above if they are anything like my two you'll need a good regime to keep their weight down if that's the only exercise they get...
Good luck!