Do not rush things. Allow suitable drying time.

Paint rollers and brushes can be wrapped tightly in saran wrap between uses and stored in the refrigerator. This way, they don't have to be cleaned and/or replaced each time. I have had rollers wrapped in saran wrap for over a week and then unwrapped them and used them again. Just cut a piece of plastic wrap of a suitable size, wrap it around the roller while it is still on the frame, then slide it off the frame, twist the ends tightly and tuck them into the tube and put it in the fridge.
For brushes, if you are going to be more than a day between uses, wash them out.

There are numerous paints and sealants which will work. Some are better, some cost more. The main thing is to use a paint or sealant which is non-toxic, washable and able to take humidity.

For the exterior, use what ever suits you. On good wood, I like to stain and then seal with polyurethane. For exterior paint, a water-based vinyl or enamel will do.

Whatever you use on the interior, let it sit for a few days to cure and set up before you put anything in it. Leave the doors off or open so that it can air out.

Sealing around the interior edges is not always necessary, but is not a bad idea. If it is a high humidity environment, it will protect the wood and will keep any excess water from seeping into adjoining cages. Also, mites can hide in the cracks, so this will make it easier to combat them should you ever have a mite infestation.
There are many paintable silicon based sealers. They usually come in tubes, so you will need the gun to hold the tubes also. Just cut off the tip at an angle, allowing the hole size to fit the size of the bead you want to lay. You can also go over the bead with your finger to smooth it out. Allow for a day or two of drying before painting.

I have used polyurethane on the insides of cages with no ill effects. It dries to a very hard surface and is easy to clean. However, if you want to have a nice color on the inside, you will want to go with paint. If you use polyurethane, figure on at least three coats. It is not cheap, but not overly expensive. Brush, rather than roller, is the method of application.

I prefer to use water-based enamel paints on the interior. First, prepare the surface by sanding. Then get all of the dust off. Then apply a coat of primer. The primer is cheaper than the paint, will seal the wood and provide better adhesion of the paint.
Use at least two coats of paint over the primer coat. Three or four won't hurt. Let the paint dry for a few hours between coats. Let the cage stand for a few days after the final coat to allow the paint to cure.

A friend from Mexico City has been working with a two-part epoxy paint which is supposed to be excellent quality and able to withstand high temperatures. This is his report on it. Thanks, JM.
The paint is working great but it's a bit difficult to work with, you need neoprene gloves, goggles and respiratory protection. It's a water based Catalyzed Epoxy from Sherwin Williams Industrial and Marine Coatings, these are the codes:

Part A		Part B
B70 W 201		B60 V 25
3886-99993		630-3994

Codes for part B are for Semi-Gloss but there's also Gloss and I guess Matte - maybe for those the codes are different. You'll know how many coats as soon as you see the first one almost dry, I use 3. For wood you'll need a primer, I don't have the codes for that one. You'll have to ask someone from Sherwin-Williams. It is best applied within a specific range for temperature and humidity. I know it sounds like a pain in the butt to use and it's a bit expensive, but the "washable, high temperature, high humidity and stain resistant hard finish" is worth it.