Weather sealing, the primary important aspect of it is dust, more than anything else.
Dust enters lenses very easily, especially zoom lenses. Think of them as vacuum chambers, as you zoom in and out. The lens extends and retracts, and to make this physical size variation easier, it need to vent air.
If water can't get into a lens that's weather sealed, then it will be a lot harder for most dust particles too as well.
I'd reckon that any of the lenses currently mentioned will be fine.
From experience tho, I think all will be a bit unwieldy on the small D3500(same as the D5500), for me they don't feel comfy at all. Not enough counter grip on the body for the right hand to help keep it on the straight and level.
On the D300 and D800(both much larger bodies) ..
The problem isn't the weight so much but the length, no matter at 150mm or 600mm, on a small body they feel far too front heavy by comparison to when fitted to a larger bodied(ie. grip) camera.
In terms of IQ, the most important aspect of this is going to be how you train yourself for it.
I'd say all these 150-600mm lenses are more like 500mm lenses. The drop off in sharpness between 500-600 can be noticeable, more so when cropped. But again, this drop in sharpness between 500 and 600mm is probably going to be less noticeable than your ability to shoot with it as you learn to use it.
If birding, some use tripods, some use monopods. When some of the best photographers use tripods you know they must work.
I just do it handheld myself, can be tiring, and I'm lazy and I generally give up easily, so when the birds are being obnoxious and anti social, I'll pull out the tripod and shoot landscapey stuff
I've mentioned this before, but the 100-400 lenses from Sigma and Tamron could also be a viable option.
From all tests I've checked they appear a bit more higher quality in terms of resolution at 400mm than the 150-600's do at 600mm, and even some of the lenses at 400mm.
More importantly they're smaller light, and this helps with IQ at long focal lengths. They're cheaper too. IQ will be better than the Nikon 70-300 by some margin too.
That would be my recommendation, maybe the Sigma 100-400(plus the USB console) .... but I reckon the Tammy version would give good results too.