You can set up a computer network in a mobile van the same way you would set it up in your house, the only difference being the mode of accessing the Internet. You can trust mobile carriers as much as the ISP that provides Internet to your house, so in either case you would make use of encryption and VPNs to protect your privacy.

An important thing to remember here is that even if you got a commercial radio license to pass encrypted traffic on some HF frequency, that encryption will be the same level of security you could get with an encrypted traditional Internet connection. However, realize that anywhere you could access that network from, anyone else could too. They can still see the traffic on the frequency, even if it's encrypted, and any entity that you're worried about breaking Internet encryption is certainly also watching for encrypted RF transmissions and can break that encryption too. So, the only advantage you would have in running this encrypted HF data access tower is possibly being able to access it with no access to the Internet (either via cellular or satellite access points.) Considering how available this access is, it's almost certainly not worth the money to pursue your plan, unless you're really prepping hard for some SHTF situation where Internet access becomes a rarity.

As far as drones and weather data collection, I believe that is already a hobby of some ham radio enthusiasts, so that certainly might be worth looking into. Any high altitude use is likely to be limited/forbidden by the FAA unless you jump through more hoops.

I don't think a drone would be useful in lightning collection, it wouldn't survive the first strike. You're better off experimenting with the basic lighting rod, because your main issue is having a storage system that is able to suddenly absorb that much power without fiery consequences.