If I'm having a conversation with someone in which I am using the polite form of verbs, should I also use the -masu form if I'm describing something I do or did? If I'm saying, for example, 私は午前八時に起きる, should that be 起きます? Or does it sound stuck-up?
At the very simplest level, regardless of whether talking about your own or others' actions, it is best to stick to ます・です style if you want to be polite.
It does not sound stuck up if the politeness is appropriate, but it might sound distant (so, potentially aloof/stuck up as a result?) if you (continually) use it when speaking to friends.
That said, it is not uncommon to have style-shift within a conversation, which is lucidly explained, in brief, here: Can polite and casual Japanese be combined?.
As is often the case with how one's words come across, it will partly depend on the manner of speaking (speed, stress, word choice, etc.), and non-verbal elements (body language, gestures, etc). I can imagine saying something in polite-form might enhance sounding stuck up in certain circumstances, but those other factors will really be key, I think.