Hi Abath-
I've seen joggers along Ela Beach and around town recently. I haven't heard much in way of problems with exercising outdoors. There are several gyms, one at the Royal Papuan Yacht Club and Lifestyle Gym by the Holiday Inn Hotel (aka the Islander, your husband may still know by the old name).
There is a pottery shop called "Annie's Pottery" who specialize in PNG style ceramics and stoneware. I'm not sure if they would have a studio for others to work at but I'm sure they can point you in the right direction.
There are quite a few clubs around. You can join the Royal Papuan Yacht Club, POM Golf, Aviat Club, All Nations Women's Group PNG Inc.T: (+0675) 71914092 E: anwgpng@yahoo.com.au, etc. You can also become a member of Moresby Arts Center, RSPCA, the list goes on.
I hope this helps!
-K