Should one  imagine diagrams/figures when working? I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw figures like : 
(Black is the set, red are the finite sub-covers)
My question is:
Am I handicapping myself by continuously drawing figures (limited to $\mathbb{R}^2$)?
I am tuned to imagine things which align neatly. For instance, if someone says imagine a triangle, I imagine an equilateral one and this usually prevents me from understanding subtle points.
 A: I think that you should only worry at what you are doing at the moment. If it helps you to vizualize something to understand it you should do it. Even when you get to higher dimensions you can still vizualize stuff in your mind. I guess you should do whatever helps you understand. Its about making it easier, not harder. 
A: The important point is that visualizing isn't a problem. If visualizations help you work faster and get more comfortable with a subject, more power to you. The issue with your example of the equilateral triangle isn't that it's visual, it's that you've chosen the simplest example possible and seem to have such tendency for simplicity/beauty. If you can remember to visualize extraneous cases and "push the boundaries" that way, then go for it. Personally I find visualizing good for gaining intuition, and then I switch to algebra for actual calculations and finding extreme cases. Visualization can be a fantastic tool, but it should just be one tool of many. It's one that many use liberally though, so don't take your preference for it as a shortcoming.
A: I can't do mathematics at all without drawing and I have practically never given a course or a talk without making coloured doodles on the blackboard.
 There are a few exceptions, as when I had to teach the Sylow theorems, but then I had the feeling that I didn't understand what I was talking about, although the proofs were (I hope!) correct in the  logical sense.
The same goes for hard analysis, say linear partial differential equations: I tried  to read Hörmander but gave up, because I couldn't get a gut feeling for the inequalities there (even though geometry is definitely present in that book).  
I am in a field (algebraic geometry) where it is easy to make drawings and I started to understand scheme theory only when I saw Mumford's drawings of $Spec (\mathbb Z)$, $Spec (\mathbb Z[T])$ and his cartoonesque rendition of the spectrum $Spec(\mathcal O_{X,O})$ of the  local ring at the origin of the plane $X=Spec(\mathbb A^2_k)$ over the field $k$, where  the closed points of  curves have disappeared and only their generic point is left behind , exactly like the grin of the Cheshire cat in Alice in Wonderland, the masterpiece of that wonderful mathematician who also loved drawings.
[If you have never seen a live scheme in its natural habitat, look at pages 72-75 of Mumford 's Red Book  where the pictures I evoke above are to be found. Or  here  pages 111-112]   
In conclusion, if you feel drawings help you, by all means go ahead: I find  your version of a covering of a compact space truly ingenious an illuminating.
  And to finish on a lighter note, there is this  story of an engineer asking a mathematician how he could visualize 4-space: "Very easy, I imagine $\mathbb R^n$ and I specialize to $n=4$"
A: The way we understand things has a lot to do with how we comprehend them. Some people are more visually inclined than others, and there is no problem with that. One learns, with time, how to "project" objects onto the mind and imagine them as shapes.
Do note that understanding how you understand things can be a key factor in finding the parts of mathematics which you may enjoy more and possible succeed in more often. For example, I actually have a problem with "low dimensional" visualization and drawings often make things harder for me. 
On the other hand I come from set theory where things are weird enough that the very little drawings have little to do with the objects (as they carry very little geometry). There are "usual" drawings in set theory but the intuition one draws from them is different, in my experience, than the intuition one draws from drawing open, closed or compact sets (the best example I saw was in the Ph.D. dissertation of Ioanna Dimitriou where she sketched how generic and symmetric extensions look like, and how a permutation of a forcing poset look like).
I think that the most important tip I can suggest is that unless you deal with specific finite objects (e.g. graphs with a tame number of vertices), drawing something is only going to take you so far with intuition. At a certain point one must return to the definitions and work with them. The key point is to be creative and figure out how to formalize what you drew on a piece of paper into pure mathematical context.
