[H]ere's another question for you guys: How likely is it I'm going to need a case trimmer for .357 and .44? I was searching through Google and found some contradicting opinions.
I trim mine for 357 & 44. I don't tend to trim pistol brass, but I do length sort my pistol brass as it comes out of the dryer, if it grows over my maximum tolerance, I put it in one bucket. Once said bucket gets full enough to warrant attention, I trim that bunch to bring it back into spec.
For blasting ammo, it's not really necessary, but it doesn't hurt. For precision shooting ammo, it does make a difference. I've even experimented with neck turning 44mag brass, and do so for all of my .357/44 B&D brass (and annealing, of course), to help get the most consistent bullet release possible.
[Is] it important to clean primer pockets every time? It seems as though the classic turret is designed to decap as part of the process, which makes me think it's not terribly important to do every time. And that the brass can be tumbled with the old primer in.
A lot of us don't use the turret presses as they were designed. I use a separate single stage press with a Lee Universal Decapping die to deprime all of my brass, then Ultrasonic clean. I have the decapping stem removed from a lot of my dies, so it only does the sizing in the press. I personally don't like to clean with primers in place, and don't like to size dirty brass.
Combining your two questions - if you're going to trim your brass, then you really won't be using the turret press for sizing, priming, charging, seating, and crimping in one uninterrupted cycle. You should trim AFTER you size, so you'll decap and size in the press, take the brass out to tumble and trim - then you'll come back to the press to prime, charge, seat, and crimp. OR - you could use a hand priming tool and charge on the bench... Or you could leave everything slack and decap and size dirty, prime, charge, seat, and crimp all without cleaning, trimming, chamfering, or deburring... Personally, I try to make very consistent ammunition, so even though I do most of my loading on a Lee Turret press, I NEVER put a dirty case in, pull 4 times, and remove a loaded round - my process is very different.