I'll have a bash at this, having completed a vaguely similar journey recently, and a regular visitor to this area over the years. I suspect that your itinerary may be a little ambitious given the time available, the risk being you end up spending too much time travelling between places and not enough enjoying them, and IMHO you will be visiting some of the most glorious scenery in the world. 3 weeks might be a more comfortable timescale if you can manage it.
Our route was SF, Yosemite (ish, we got as far as Sonora, where alcohol did bad things to us), Barstow (I-40 south of Death Valley), Vegas and Lake Powell (looping south on 89A rather than north through Zion and Bryce), Monument Valley.
Although this doesn't take in all the sites you mention the distances work out very roughly similar, maybe even shorter. Just to drive this route, and get to Phoenix I would estimate (based on our experience, no more than 300-400 miles per day, tiring but not pushing it) at around a week. Depending where you're from you might have jetlag to consider in all of this. Yosemite is the only destination you list that I haven't been to, and assuming you have no desire to sight see your start and end points (though if you do I think SF is, probably quite obviously, the better choice) that's 6 of some of best that the US has to offer, 8 if you try and squeeze in both Zion and Bryce as well as Monument Valley, which doesn't leave a lot of time to enjoy the sites once you get there.
Recommendations can only ever be a personal thing (me - I love desert and heat), but if I were planning such a trip (based inevitably on hindsight) I would...
Fly into Vegas rather than SF and enjoy the bright lights and tack there for at least two nights (allow for sleeping it off)
Loop north to take in Zion (perhaps a day to drive there, a day to see it, staying a second night to get the full day there - I've stopped in Springdale before now).
On to Bryce, maybe squeezing in the journey there, a quick visit and moving on.
From Bryce to Lake Powell via Kanab and Fredonia so that you can pick up the 89A which is my favourite scenic route (Vermillion Cliffs and a nice view down onto the Colorado as it cuts into the desert from the road as it climbs the escarpment before descending into Page). Maybe an overnight stop on the way, maybe push all the way on. Allow two days at Page/Lake Powell so you can spend a day on the lake (boats are expensive but magic, but see options below).
On to Monument Valley, probably a day's drive there, so one night at Gouldings right next to MV, or maybe in Kayenta, 23 miles south, for a leisurely day exploring the Mesas and Buttes at MV. One of the great attractions of MV (after the scenery) for me is the fact that it is a Navajo tribal park rather than a US national park. The latter are brilliant as far as I am concerned, but there is something deeply profound about Native America, and a friendlier bunch of people it is difficult to find (in a country that I generally find to be friendly anyway).
On the return journey via Page/Lake Powell take the 89 south, and either the 64 west if you can stay in a Grand Canyon lodge or find accommodation in nearby Tusayan (pricey), or keep going south on 89 and then I-40 west to stay in Williams, either probably a good days driving (accommodation in the GC is often booked up well in advance, though I could have had the last couple of rooms at a few weeks notice in one of the lodges on the rim's edge one Feb if I wanted to pay the price - this was mid 90's. I've always stayed in Williams, about an hour's drive(?) south of the south rim on I-40 when I've visited the GC). Perhaps a day to explore the Grand Canyon.
With a day's drive to Phoenix that's about 12 days of travelling and speed sight seeing. I'm a bit hazy on actual travelling times and distances so do try and check these for yourself.
If I were to give up any of these in order to spend more time elsewhere it would be...
Grand Canyon. It's great to say you have been there, and wonderful in its own right, but compared to the other destinations listed I found it slightly disappointing. I've never visited the north rim which, I believ is closed over winter (though when excatly winter ends there I'm not sure), and less touristy (you'll certainly be mixing it up with the tourists at the south rim).
Lake Powell. Still my favourite place on the planet so far, but falling water levels after years of drought leave watermarks on the rock which detract from the scenery a little. Also the sheer joy of this place is spending a day on the lake, beached up in a private little cove, surrounded by Orange rock and basking under a blazingly hot sun. For me it just wouldn't be bakingly hot enough in April.
Monument Valley. Not that it's not worth seeing, it really is (again, IMHO), but it is a relatively time consuming journey. If you have the money see if you can get a sight seeing flight out of Page airport that will take you over the Valley (you'll also get to see the Lake Powell area on such a trip as well).
One more bit of advice I can offer. Although this whole area is desert country, I've been to the Grand Canyon in Feb and it was jolly cold. I can't remember what time of year I visited Bryce, but at 8000ft above sea level and higher than the GC (IIRC) it too can be quite cold at the wrong time of year.
Death Valley is always worth a visit, but it's in the opposite direction of so many other great places if you're starting out from Vegas. If you're coming in from SF and Yosemite it's worth taking in, but you may end up sacrificing everything east of Bryce with such an itinerary.
Even with this itinerary you'll miss the Painted Desert (don't really recall seeing this, though it's on the route south from Page/Lake Powell), Sedona (beautiful little hippy town south of Flagstaff) and more I'm sure.
Whatever you end up doing I hope you blog this trip. I've been to this area at least five times, and I do not think I can ever tire of it.
I hope this has been helpful.
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