What cage is she currently in?
Because yes pet shop stuff are fine to extent but usually not cage wise
Yes though, they do learn that by bar chewing they get attention, but lots of syrians are like that so don't worry. I guess you've just got to get her out more!
Tubey cages.. uhm nobody on this forum would recommend rotastak or habitrail to be honest. Pick out any rotastak review on here and you'll see why. And yes, chunky syrians probably will have difficulty getting through the tubes. If you have a look on equinecaninefeline or zooplus then there are some nice cages on there, for example the alaska on zooplus for £30