I did that exact trip (Wire Pass to Lee's Ferry, in four days) last year in the exact same week (first week of June). We used Yermo for the shuttle and I'd recommend him. I agree that starting at Wire Pass makes more sense than starting at Lee's Ferry; if weather turns on you, you've done the best part (Buckskin the top parts of Paria) and can retreat to White House (whereas the last day is much worse and you wouldn't want to do an in-and-out on that part). I bet it's also less tiring to walk downstream since you're going with the current.
When we did Buckskin it was bone dry, but it completely depends on the year, and in particular, the recent weather. The nice thing is, you can just call the ranger station or Yermo and either of them probably knows the current conditions. So Buckskin was super easy for us (just flat walking the whole way), although the sand got a little tiring. We did the shuttle the evening before and camped by the trailhead, and I recommend doing likewise to enable an early start (so you can take your time on the long first day). I know in some years there can be extensive pools, and sometimes they're quite gross/stagnant. In any case, Buckskin Gulch is one of the most spectacular places you'll ever go so you'll love it! Buckskin is all just flat walking (no scrambling whatsoever IIRC) except for the boulder jam, which was harder than I expected it to be. There were a couple of ropes there to help you down that, and we sure couldn't have done it without them! I think the rangers and/or Yermo will probably also know the state of the boulder jam ropes.
The first week of June was very hot outside of the canyons, but the first three days it was surprisingly easy to stay cool. Day 1 (Buckskin) is like 90% in the shade and way cooler than in the sun. It was surprisingly nice and cool the whole first day. Then the next two days there was water the whole time, so we frequently got in the water to cool off (just like Janice shows in her photo). I hate the heat in general but the first three days ended up being pleasant overall. The last day was very hot, so we woke up early to beat the heat as much as possible. We wore shorts (my first time doing so while backpacking in years!) and that was the right call, although we had some biting flies the last day or two (pants still would have been worse the last day, because of the heat--the flies weren't that bad). I actually brought a sun umbrella for the last day (my first time ever doing so), and I think it was worth carrying the whole trip just to use the last several hours. If you can't tell, I hate the heat though. We brought some electrolyte powder to mix with water (again, something I don't normally do) and I think that was helpful as well.
I don't think we had any mosquitos, and there weren't any cedar gnats. Of course that just depends on the year though. There was some quicksand but not much (my son sunk about to his knees once but that's the only time I remember). There are mice issues at the popular campsites near the confluence of Buckskin/Paria. I had one Ursack and one stuff sack, and even though I hung them on a rope off the cliff wall, the mice still chewed into the stuff sack and got into my food. If I only brought Ursacks, they might have chewed through those too. I would recommend some kind of mouse-resistant container for the first night.
The Paria was surprisingly pristine for such a popular canyon. The permits and WAG bag requirements are really working. We saw hardly anyone for the last 3.5 days (we went mid-week, and literally saw like one group per day for the last 3.5 days), and once we got out of Buckskin, we didn't see any trash/TP or any other signs of humans other than established campsites. Before the trip, I was wondering if WAG bags would really be necessary in the lower parts of the Paria (and was thinking toward the end of the trip I might go way up side drainages and dig a hole), but the way the canyon is, the WAG bags really are necessary and I would plan on using them through the entire trip. Totally worth it to keep a canyon like that pristine.
I don't remember where exactly we camped the 2nd night, but it was a little past the Judd Hollow pump if I recall correctly. I think there were a few established campsite options in that stretch and not much competition for them (at least when we went). A lot of the people just do Buckskin to White House so once you head down the Paria from there you probably won't have campsite issues (I actually think the 2nd night is the easiest to find a campsite).
Definitely do Wrather Canyon. I thought the canyon itself was spectacular, even if there weren't a massive arch at the end. I also recommend heading a little up the Paria from the Confluence as that first little bit (1/2 mile?) is spectacular. I also recommend checking out Shower Spring. It was actually not obvious where it was, and we had to poke around a little to find it, but it turned into a memorable adventure and was a great spring. Also, on the last day, I would budget some time to look around for the petroglyphs on the boulders. Keep your eyes peeled and you will see that the trail goes right by a boulder with a bunch of petroglyphs on it (I think you might have to turn around to see them--they're facing the other direction I think). Once you see that boulder, go exploring on the hills around there for an hour or so and you will find many more such boulders!
One more thing--the restaurant in Cliff Dwellers tasted great before and after the trip for us! Not bad for being the only thing around. Have fun!
Here's two photos of the boulder jam (there were two possible ways down):
