Yeah climbing shoes are recommended. If you are keen about climbing get a pair. Go speak to steve at Rock Hardware - Neale st (near McIvor Highway) and he will get you a good fitting pair of shoes. Have a chat to him about what you want to use them for and you should be sweet. He is super friendly and has heaps of experience and advice and will give you a good deal for them.
In regards to the Uni wall you would be more than welcome to use it. All the climbers at the uni are friendly and stoked to meet keen climbers. Just say Hi to anyone that you see and they will show you some of the climbs and give you some tips. Most of the time there is no one there so just jump on it and have some fun. The over hanging wall is a bit pumpy at first but after a while it becomes really fun. Great for bouldering and endurance training.