Should I leave the water bottle tip open or shut while in the cage?
To quote David Richerby: You're over-thinking this.

Most people close their bottles most of the time.

I'm not one to worry about a little "gunk", but I close my bottle more than half the time. I almost never wipe it (heh, I almost never wash it :-)

Not fanatical about it though. If I'm taking a quick swig before a corner then I'll just jamb the bottle back, whether the top's open or closed, before I get to the corner.

That can happen a lot, because I drink a little, often.