How long the doxy will last in water is dependent on its form. The monohydrate lasts longer than the hyclate for example, which can start to degrade after a day or two: (first two paragraphs; Veterinary Drug Handbook, 2008)
When you're dealing with powder its best to get a scale capable of measuring down to the mg, such as this one:
https://www.amazon.com/gp/product/B00ESHDGOI
Of course for $20 don't expect it to be able to actually measure accurately and repeatably down to the milligram- a scale that can do that will cost 10x as much. It will however do a great job measuring in 10mg increments which is plenty for your needs to make a solution from powder so you can dose accurately, without wasting too much.
Whenever you're diluting drugs for rats one important thing to remember is you want the minimum volume you can get away with for any given dose. It's a lot easier to feed them 0.2ml than it is 2ml. A lot of times you can get the dosage down and then mix it with yogurt or something else they like. If you keep the volume of diluent to a minimum they will often not even notice.
For example the vet always tries to sell you the big bottle of Clamavox powder with water already added. They don't trust people to be able to weigh powder, so they mix it up at the time of your visit and then the bottle tells you to dispose after 10 days or so. I always insist they not add the water, because not only does that start the clock on the shelf life (the powder will last years if you don't add water), but it means I have to give them a syringe full of flavored liquid with a flavor they may hate (and the dosage volume is formulated for cats and dogs, not rats) rather than dissolving the powder in the absolute minimum quantity of strawberry ensure which masks the taste very well and is much easier to syringe feed to a rat.
If you're interested- after I weight the powder, I put it in
a microcentrifuge tube, and then add the ensure, and then mix with a
vortex mixer. The small tube means you don't lose much left stuck to the walls of the mixing container. You could just shake it by hand of course, but the mixer keeps bubbles from forming and it's super useful for other things too.