Unlike most other brewers, personally I never use homegrown hops to dry hop. Too many questions and issues with dunking a bag in there, hop particles, etc. Plus I have experienced a contaminated batch from trying this. It might work better if you weigh the hops down with sanitized marbles or something to keep the hops and bag fully submerged.
With my homegrown hops, I like to aim to make a pale ale. A simple pale ale with extract is just a few pounds of light DME, with ~15% crystal 40 or 60 or whatever you like, with US-05 yeast. I use my homegrown hops for bittering, flavor and aroma, adding them at 60 minutes left in the boil, then 15 minutes and 2 minutes. This works great as long as you are flexible about what style you call it. If it turns out right, great, it’s a pale ale. Turned out too weak and not very bitter? Fine, so then call it a blonde ale or amber ale, depending on color. Way too bitter? Great, then it’s an IPA. No big deal, as an experiment. They all taste great regardless.
If alpha acid matters to you and you have the capability of making more than one batch with your hops, then be sure to jot down a guess as to the alpha acid content and how much you used, and enter this into software. Because then if it tastes way too bitter or not bitter enough, you can take a second guess on the alpha acid % for next time, and in this way you can determine very closely after a couple of batches how much alpha acid content there was.
As an example, I find that for my homegrown Hallertau hops, the alpha acid has ranged from about 3 to 7% in different years. This has a lot to do with how much water they got, and how late you pick. Generally the later you harvest, the higher the alpha acid, but if there was a drought in the summer then this can hurt it, etc.
I would go with the half batch, at 2.5 gallons, or even a 1/3 size batch at 1.67 gallons. Then if it turns out great, you can brew again but with greater knowledge of your bitterness and flavor compounds. If it doesn’t turn out so well, it’s less to swallow or dump! A one-gallon batch is not the best size in my opinion as you’ll get just a 6-pack for all your labors… but if that’s good enough for you then it’s good enough for me! Just means you have more hops to run extra experiments!
You shouldn’t really need any special equipment for this. Just use your normal fermenter and everything and you’ll be fine. You can chill your wort by immersion in a sink of cold water, replacing the immersion water after about 20 minutes after it warms up.