Recoil of a spring air rifle is forward. If the scope is designed for rearward recoil, it will come apart internally very quickly. I placed a 1x4 scope on a pump Mossberg 500 and destroyed the scope in two rounds, by pumping forward too hard.

Potentially Yes

The recoil impulse is very different.
It’s not about the push of energy kicking you in the shoulder, it’s more about the sharp FWD metal to metal harmonic reverberation that goes through the optic.

I spent the last four days researching scopes and it will make your head spin.

Some questions for you and comments:
- what gun are you mounting the scope onto? If it's a PCP or pneumatically charged gun, you might not have as big of an issue like you will on a springer.
- lets assume springer. Do you know the energy of the gun, or, ft/lb engery (FPE)?  I have a Beeman R7 .177 and the energy at the barrel is 4-7 FPE depending on pellet. I also have a Weihrauch HW97 .22 with muzzle energy of 14/16 FPE.
- What is your budget? Include $20/$80 for rings or one-piece mount.
- So, I needed to run a fully dedicated springer scope on the HW97 (Hawke Airmax 4-12x40) AMX but chose the Hawke 2-7x32 Vantage for the R7.
- I also bought a really nice mount for the HW97 which is reported to have dampening built in. Sportmatch DM-60 Damps ($72)
- Googling will give you some paths to follow depending on what gun you have (from people a lot smarter than me). Look for reviews and input from sites like Airgun Nation and Gateway to Airguns.

Some stuff I learned?
- Leupold and Nikon (when they still made them) with EFR rating can stand up to springers
- Hawke Airmax line, but there are some haters out there
- UTG Leapers, did not like the "36 color" lighted reticle and is reported to have  heavy/thick recital lines
- This scope and mount gets a ton of positive reviews for holding up to magnum (17 FPE+) air rifle. I couldn't buy in but it's only $80 and a 4-12x40 Here on Amazon
- You will read a lot of "I heard Burris, BSA, Vortex, ect" are springer rated... Heard and verified are two different things.
- Lots of love for SWFA fixed zoom scopes. Fixed = less moving parts to subject to spring recoil.
- Read up on First Focal Plane and Second Focal Plane. I went SFP. Good sites explain what it is and what you might want. SFP is more common but FFP is catching on. All about sight picture and if your hunting, target, etc.
- Probably want a scope that focuses down to 10m (yards)
- There are a ton more, it's a rabbit hole

Others with more knowledge will chime in, hope this helps.

now get yourself a homemade pellet trap and see what kind of groups it shoots.

Look up "artillery hold for springer". It shows how to hold the front stock on a springer for best accuracy.
here is one

Reading about "red" dots over at Airgun Nation, many have used them with no issues. I think your fine.
I'll add, if your red dot is a Weaver set up, you will need a 11mm dovetail to Weaver adaptor.

thread here

Yes...
you could go with any length and they are available everywhere (Amazon)

I hope it warms up for a range report.

That should work for vermin. Make a quick bag rest out of rice inside a one-gallon Ziplock, then put that in a fabric bag. Works pretty well unless you have a dedicated rest bag.

Good shooting.

Perfect... any European Starlings around? If so, hang a suit block up and have at it.

If you can get the BB tip gamo pellets they penetrate like crazy. I have shot several skunks with my RWS 34 in .177.