You shouldn't need your own horse. Talk to your instructor about it :)
I would talk to your coach and she what they say.
No you should be able to use your barns horses. Ask your trainer if you have one
It really depends on your barn's policy. Some barns will let you show lesson horses but some won't. If you're planning on showing frequently I recommend leasing or owning.
not just to show but if you want to be competitive leasing and or owning is better because you get to know the horse and have a better "bond" .
You can talk to the head of the centre and ask if you can take their horse to the show. There may be a small charge to lease the horse for the day