When you click on links to various merchants on this site and make a purchase, this can result in this site earning a commission. Affiliate programs and affiliations include, but are not limited to, the eBay Partner Network.
If you leave it as is it will get dirt and moisture in it and yes it will rust. If you don't want to replace the antenna just cut a small piece of black electrical tape and place over the hole to keep it clean.
It looks like they might have gotten a little rough. There was a small space between the car top and the base rubber on the mount. I smacked down on it and it looks like its seated ok.
Personally, I would get it checked by the dealer. You really don't water getting in there and ruining your headliner and whatever else is up in the roof.
I have a stock antenna in perfect condition for sale if you want to buy it? -PM me if your interested (I replaced my stock one w/ a stubby antenna in the first couple months after owning my Mini). I'll sell it for $15 -this will include shipping and packaging.