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How Delicate is it?

That’s going to vary from gun to gun. In your example, I’d be more concerned about altering my point of impact (POI) by bumping a tensioned (I meant a) free floating barrel, falling with it, or leaning it against trees or walls as opposed to a barrel enclosed by a thick and sturdy shroud as seen on a Benjamin Bulldog. Some airguns are marketed as rugged and tough guns that can stand up to rough treatment. For example, I’ve read that EDguns were advertised as being tough enough to withstand being run over by a truck. Good question, but again the answer will vary from gun to gun. You may wanna ask guys like @crosman999 or @davidsng for more details based upon experience with a lot of guns. @Crosman999 hunts a lot and says he’s hard on some of his rifles. What I’m curious about are the risks of hunting in the rain with an airgun. I’ve been wondering if that might adversely affect air rifles. 
 
That’s a fairly loaded question. There are rifles that are quite delicate then others that can be ran over with a truck with no change in POI. Generally rifles without a barrel band ( free floating ) tend to be on the delicate side with some exceptions of course. I tend to stick with ones that have shrouded barrels as I find this gives them a bit of extra strength as well as having a barrel band of sorts. Others are built where the receiver it screwed down to the tube ( discovery , hatsan AT44/BT65 ) this is not what I would call a solid design. I again prefer models that have a block design where the tube/bottle and barrel are threaded into or slip fit into the receiver. Like RAW, Evanix, Kral , FX crown etc. which are in my opinion a much more solid design. 
 
All floating barrel guns will be “delicate”, all tensioned and shrouded guns will be more rugged. All precision guns will be “delicates” and you won’t see anyone who shoot bench rest throw their guns on the ground let along run it over with anything. Different tool for different jobs, you don’t drive Ferrari on dirt and F150 won’t win any track races either. 
 
Yeah, it all depends on the specific rifle. While rarely the most robust, some free floated barrels are far more sturdy than "delicate". If the barrel shank is relatively long, and held securely, it can take a healthy bump without issue. I consider RAW such a rifle, although it is not a handy carrying rifle for me. The problem with some rifles that use forward support, the piece furnishing the support can move with a bump, thus preventing the barrel from returning to zero. 

If you are considering a PCP for hunting, you could put the question on that forum for some practical feedback.
 
Sounds like you are looking for a robust gun that can take its bumps and keep on going. Based on my experience with them, I would take a look at the following brands:

  • American Air Arms (AAA) Evol - this rifle is so robust I think you could just walk up to your game, beat it to death, and then shoot the next one no problem. 
  • EdGun - ran over his own rifle with a truck and then shot 50 and 100 meter targets with it - 'nough said.
  • Kabibrgun - My Cricket II is very robust and I can't think of any woods treking that would be able to impact its POI.
  • Taipan Veteran - very robust build, should handle a few bumps with no issue

I'm sure there are plenty others, but those are ones I've owned and spent time with.