Would a refurbished Macbook Air have sufficient computing capacity to meet my needs?
2009 was the switch from Core2Duo to the Core i3/i5/i7 range. In short the new laptop should walk all over the Core2Duo, but if it’s a newer model it might just be a little more evenly matched. But in short for the requirements you have listed a MacBook Air would be more than enough.
Would the non retina screen be a real factor?
That’s more up to you. Personally I never had an issue with the MBA screen and although my retina screen is nice I don’t think I would have a real issue (with the non retina-ness) if I was forced to go back to a MBA.
Which leads nicely into: Would I really notice the change to a laptop?
YES. Screen size and total resolution here are huge factors. Even going from a 21.5" down to a 13" laptop is going to be huge. I love my screen realestate and run my retina 15" MBP at 1920 x 1200 so I get more to work with. Having less space to work with, even going from 15" retina to 13" non retina would hurt alot. I’m sure I could get used to it, but that would be my biggest issue.
You can always add in a keyboard, mouse and external screen to give you the big screen experience as well as the laptop portability. If you’re happy with 22"/24" screens these are pretty damn cheap, especially if you grab one off gumtree or similar.
Honestly, if you don’t need the laptop portability a refurbed iMac might be the better option, although I just went to look up pricing for reburbs and there it literally nothing in the mac refurb store right now. It’s also sort of hard to compare laptops to the iMacs given the way they provision SSD and/or fusion drives makes pricing pretty different… also modern iMacs are hard to upgrade yourself after the fact.