I use paratroopers extensively; they're my go-to strategy in most campaigns.
I've found that a lot of these weird, unexplained "freezes" in para orders are related to air zones. If your target province lies within an air zone whose central node is outside the range of your transport planes, then it can cause an infinite delay and prevent paradrops. What this means is that, in a number of airfields, provinces might technically be in range of your transport planes, but your planes will never arrive at those provinces, because the target air zone's coverage node is outside of their range (or something like that). The Germany-to-Southern-England thing is a particularly obnoxious example, but it happens in a lot of other places too (the Pacific and South America, for example).
Fortunately, you can get around this broken game mechanic by, well, breaking it further. When your para drop order appears on the Air map screen (same place where you assign Air Superiority and Ground Support missions), left-click the order's circular icon (probably looks blank), then right-click reassign your order to an air zone whose central node is closer to your airfield (this will usually, but not always, be the air zone your airfield is located in).
Paras should drop within a few hours.
There's other problems, too - for example, if supply gets severely into the red, if you try editing the paradrop order on your Army screen more than an hour or two after issuing it, and if you save/reload after issuing a paradrop order but before your assigned division actually drops, all of these things can cause hangs (or worse!). But, in my experience, unexplained problems with paradrops ("I've got air coverage, I've got the right division template, etc etc") are almost always related to air zone coverage and the range of your transports.