I can't answer any of your questions on PEX. I don't have any experience with them. All the work I've done around my house has been with copper and PVC.
First, for backflow preventor, I assume when you are talking about a double check valve, you really mean a double check valve backflow preventer. Logically, they are SORT of the same thing, except the backflow preventor will have the needed test cocks that can be used to prove each check valve is properly working independantly of the other. Simply putting in a double check valve is NOT considered proper backflow prevension.
Second, are double check backflow preventers legal where you live for irrigation backflow prevention. Local codes vary greatly on this issue, with some being so strict that allowed backflow preventers are specified by part numbers and required a certified plumber to install them... to some simply saying something along the lines of "adiquate backflow prevension is required".
If a double check backflow preventor is legal where you live, and you never plan to inject anything into the irrigation lines (such a fertigation equipment to fertilize as you irrigate) then YES, a double check backflow preventor can be installed in the basement, or just about any where in any orientation. Unless specified otherwise by local codes, the only limitation is that you must have it installed such that the test cocks are properly accessable for testing the device.
Given that you are talking about 1" pipes running underground from the manifold to the planting beds (at which time you are transitioning to 1/2" drip tubing for above ground), the flow rates should be relatively so low that you shouldn't have to worry about pressure losses through the 1" pipe and therefore you can just install a single pressure regulator at the valve.
One of the things you seem to have left out is a filter. Because of the tiny pores associated with drip equipment, drip irrigation REQUIRES that the wate be filtered with a very fine mesh filter. If you get a drip zone assymbly (something like one of these:
http://www.sprinklerwarehouse.com/Control-Zone-Kits-s/134.htm), the filter will come with the valve and regulator.
Otherwise, the only other thing I can mention is you have to make sure you are following codes for pipe going from the point that you tie into your water main to the point of the irrigation valves. That pipe will be under constant pressure and therefore you have to make sure the pipe is suitable for that. In my case with copper and PVC, PVC is ONLY rated for main line water pipes when they are buried 18" underground (mainline in this context means pipes under constant pressure, while the pipes AFTER the irrigation valves are lateral lines and are only pressurized when the zone is on and therefore do not have to follow the same standards as mainline pipes). So for me, I would have to tie into my copper mainline in the basement, use copper to the backflow preventer, and then copper out of the basement.
However, I instead tied into my mainline out at the street where is was a 1" burried PVC.
If you install a connection point to blow out the system for winterization, such a connection must be AFTER the backflow preventor, and you should install a shutoff between the backflow and the blowout port. (The backflow preventer likely has a shutoff and the end of it that could be used to block pressurize air from heading back toward the backflow during blowouts... but since you are just installing stuff, adding an additional shutoff allows you to isolate the backflow preventor should repairs ever be needed on it. And of course you need another shutoff between your mainline and the backflow preventor.