I always leave my valves open, and never have any issue with anything getting in. The water bottles I use have fairly enclosed orifices and it would have to be quite the dust storm for anything to get in.

I have no trouble retrieving a water bottle and getting a drink while riding, but I still nearly always find myself coasting while actually drinking. Sometimes I make myself keep pedaling, but it doesn't come naturally. For whatever reason, pausing the pedaling while drinking is my instinct, though that wouldn't go over too well in a group ride if we were pacelining.

To the OP: it will help a lot if you can get used to grabbing your bottle and putting it back without looking at it. I just reach down and by muscle memory I can grab it. Sometimes I'll take a quick glance down to orient myself before putting it back after drinking, but still put it back while not watching it, since I prefer to keep my eyes on the road. One little problem with my approach is I find it much easier to retrieve and use my front water bottle than my rear bottle (the one mounted to the seat tube), so usually when the first bottle is empty I'll go through this awkward exchange where I swap the two bottles so I'm always drinking from the one mounted to the downtube. I'd second what the others have said and just practice on your own retrieving and putting your bottle back while riding, pausing pedaling while doing so if you need to. Once you've got that down if your valves are left open you can just squirt some water in your mouth. I usually have an inch or so of water left in my bottles when they're "empty" according to my usual practice, because I prefer to squirt water into my mouth rather than suck water through the valve, because I don't have to tilt my head to sky to do it. If I still need water and both bottles are down to that last inch I'll tilt my head and drain it.