Thanks Lisa! Hi Niels, I've sent you an email about the owls and my guiding services.
To answer the other questions...
Some sites are best reached by a high clearance vehicle, but 4x4 is rarely required as it's usually dry and rocky conditions you're dealing with, not slippery mud. The only time I ever got stuck was in sand, and that was in a 4x4 anyway.
The main site that is best by high clearance vehicle is California Gulch, although the road is much better than it used to be and can be accessed with care in any vehicle. It is quite rough, but if you're brave enough you can do it in a regular car. It helps when it's not your car!
***IMPORTANT*** It should be noted that this is not the case in the monsoon season (July-August) or after it has rained, as some of the dirt roads get very muddy, especially in the valleys in winter. During the monsoon season, California Gulch, and one or two other sites, can be completely cut off. It is never safe to cross fast flowing water during the monsoon season, flash floods kill people every year (not normally birders!)
Other sites that definitely require high clearance include Chino Canyon, French Joe Canyon and Rock Corral Canyon, plus some of the more obscure places you can find in SE Arizona.
On to the other birds... Williamson's Sapsucker is quite rare in SE Arizona even in winter, and very unlikely in May. Likewise, there's no chance of American Three-toed Woodpecker in SE Arizona (never recorded) but it does occur in Northern Arizona, where it is also rare and hard to find.
As mentioned above, Varied Bunting typically arrives in the third week of May, or thereabouts, so you might get lucky with one of the first returning birds. Most of them are down south where I live in Santa Cruz County - I'm lucky enough to have it as a yard bird! Sites to check include California Gulch, Peña Blanca Lake/Canyon area, Kino Springs, Patagonia Lake area, and Patagonia Roadside Rest. They also occur as far north as Tucson, but the highest concentration is towards the border. They can be quite common and easy in the right habitat, but not usually until June.
Five-striped Sparrows occur regularly at a few sites, but none of them are easy to get to. Chino Canyon has a few pairs but the road is horrible and I've never dared take my 4x4 SUV in there because I need it for work. Montosa Canyon occasionally gets a pair or two, but isn't reliable. There's a population at the bottom of Sycamore Canyon, but that's a brutal six miles of hiking/climbing and I wouldn't recommend it (I've never got further than a particularly tricky spot of rock climbing and rope swinging two miles along).
That leaves California Gulch, which is by far the best place to see them. See note above about access. Finding the right spot is the hardest part, so follow the instructions carefully. It's easier than it used to be as the roads are vaguely signposted now, and it's easier to tell which is the 'main' dirt road than it used to be.
The classic hike down Heartbreak Hill will give you access to the main gulch, where there were at least 20 pairs last year. 2015 was a good year for the sparrows, and I saw several pairs further south, at the confluence of California Gulch and Warsaw Canyon, and even a couple of pairs had spilled over the ridge into Warsaw Canyon itself.
The sparrows are probably mostly resident (despite what it says in the field guides) but extremely hard to find until the breeding season. They don't nest until just before the monsoon, timing the hatching of their young with the abundance of insects the monsoon brings in July and August (like Botteri's and Cassin's Sparrows). From May onward they become increasingly easy. By July they can appear to be the commonest bird in the canyon, with males up and singing away. Knowing the songs and calls is important.
California Gulch is a great place, and I go there frequently with my clients. I highly recommend it, but you do have to take it seriously from a safety point of view. It's remote, with no cell phone service, it's known as a smuggling and immigration route (although in reality this has never caused me a problem) and the flash flood risk is very real. Make sure you have a good spare tire, plenty of water and a full tank of gas. Increased numbers of Border Patrol agents make it safer, so attract their attention if you have any problems.
Good luck!