Yes, you should closely match your batch size to the carboy. So, either make three gallons and use a three gallon carboy, or buy two one-gallon jugs. (I use Carlo Rossi wine jugs- a #6 stopper and airlock fit great!).
In primary, you can use a bucket or a larger carboy because oxygen is good for the yeast at that time. But after the fermentation slows, and you rack to secondary, you want to have the absolute minimum of headspace possible. Oxygen is the death of wine after that. It'll turn brownish and taste like sherry if it becomes oxidized.
I'd buy some racking tubing (so you can siphon to a new container several times during this process), and make sure you have a good quality sanitizer. You can use potassium metabisulfite for this, but I like to use a no-rinse sanitizer like one-step or star-san.
Just about any wine yeast will take the wine dry- that's what it does. That's fine. After it's completely clear and finished, you "stabilize" the wine with campden tablets and potassium sorbate (or "wine conditioner" is sold, as well) and then sweeten to taste before bottling.
For specific strains of yeast, I would probably use Red Star's cotes de blancs, or Lalvin's k1-v1116 (montpelier).
Here's some recipes from my favorite winemaking site:
winemaking: Requested Recipe (Strawberry Wines)
I really like his common sense way of explaining basic winemaking:
winemaking: The Basic Steps
Strawberry wine is a very nice wine, and pretty simple and straightforward. You should have no problem with it!