A 5-gallon nursery pot is a decent size for a mature 3 fan division, and with proper culture you can see nice scapes & flowering for a couple of seasons, but don't expect the kind of heavy, robust growth you'd see in a garden bed. If you like to trade and share plants, you'll probably see better multiplication in a container, probably due to higher soil temperatures, and it's easier to divide container-grown plants.
If you want to see container plants really develop, a 15-20 gallon tree pot is best. Check with some topsoil yards for a fairly open mix designed for palm trees in large containers, because buying bagged soil mixes from nursery centers becomes ridiculously expensive. Note that a topsoil-based mix might not do well in 5-gallon or smaller containers.
With your heat, you'll probably need to arrange your pots to keep the sun off of the sides. Warmer soil is good, to a point, but the sun will cook any roots which are near the side of a black plastic pot.
I plant most new arrivals in a pot that I feel is proportional to the size of the root system I receive. This might be a 1-gallon pot for a mail-order single fan. If the roots are large, I'll go bigger. After the plant grows to the point where there are enough roots to hold that soil together, I'll bump up to the next size, and so on until I hit 5-gallon size. At that point, further re-potting is a matter of discarding extra fans, because I have very few containers which are larger. I like stepping them up because I think they develop better when given a layer of fresh potting soil as they grow. Also, potting soil "ages-out", and potting up in stages seems to delay that inevitability. It's also very fast/simple to step-up a size; just put some fresh soil in the bottom of the new pot, set the undisturbed root ball onto it, and fill around the sides. If the daylily has pulled itself down too deep in the first pot, crumble away the top of the root ball, and add more soil to the pot so that the crown sits where you want it. I like to place the fans pretty high, and if the roots are still covered after watering in, that's deep enough for me. Shallow planting seems to help with increase. I feel that simply plunking a newly-acquired single or double fan plant into a 5-gallon container is counterproductive, since the roots will quickly reach the sides and start to circle. Once a root is running along the side of a pot it's really only half of a root, and the plant will lose vigor.
. I'm not very consistent about the potting mix. Using potting soil of any brand is too expensive for large pots. I tend to pad it with pine bark. The finer the better, but it's hard to come by so I use larger often. I also use anything else I have on hand - like oak leaves that been put in a compost container and haven't quite turned to compost yet. Anything to add volume and reduce the amount of potting soil. That has also included fresh wood chips the power line people left in exchange for place to park their equipment. Surprisingly, using that ground up wood didn't lead to as much yellow foliage as I was expecting. On some plants it seemed to have no effect at all and a lot of those that did turn yellow seemed to outgrow it as the temperatures rose during the summer. Keeping a potted container with sufficient moisture requires regular attention and plants in a container have their roots more exposed to high and low temperatures than when they are in the ground.
