What is the estimated original gravity for the recipe?
The fermentation should take no longer than a moderate gravity ale with good aeration of the cooled wort in the fermentor and pitching the right amount of yeast. I usually dry hop around day 14 from the start of the fermentation. By day 21 the hop pellets have dropped and the beer has cleared. Timing will depend on specific gravity to determine when the fermentation is complete and dry hopping can proceed.
I use this calculator to determine how much yeast to pitch.
Alternatively one 11 gram pack of rehydrated dry yeast is good for five gallons of ale up to an OG of 1.060. One-half of an 11 gram pack should be more than enough for your 1 gallon batch even if it is well over 1.060.
The stick on thermometer strips are good for tracking the temperature of the fermenting beer. You will need some kind of temperature control if the area you are fermenting in is room temperature. A water bath can absorb some of the heat produced by the yeast as the wort ferments. A wet towel around the fermentor can add additional cooling by evaporation. A fan blowing on the wet towel can accelerate the cooling effect of the swamp cooler.
Don't submerge thermometer strips if using a water bath.
I hope others will add to this. I've only done moderate gravity IPAs.