1. Are any of the non-Apple products compatible with the AE and would work here?
None that I could recommend. You can use any universal repeater.. but the results are no where near as good as sticking to one brand.
The exception is to use Powerline products. e.g. TP-Link AV1200 with any WAP at the other end.. it will be more expensive than another AE but could perform a lot better. There are also combined models with powerline plus wireless.
TL-WPA8630P KIT | AV1200 Gigabit Passthrough Powerline ac Wi-Fi Kit | TP-Link
I am assuming you are in the US but they are made with EU, UK, AUS, etc power sockets.
2. Could I use an Apple Express instead? Does the Airport Extreme offer any advantages as an extender compared to the Express?
If you currently have AC model extreme they are hugely superior to the Express for extending wireless.. better range, much faster wireless and full gigabit ports. Express has one big advantage.. airplay.. otherwise it is old N wireless, slower and poorer range and 10/100 ports.
3. Would I be better offer just getting a whole new system with a new router? However, I do like the AE since I'm not very network savvy and it seems user friendly.
If you have a pair of AC extremes.. then buy another one.. second hand they should go for $100.. at least where I live we get them for that price now. If you have a pair of Gen5 (A1408) or earlier or mixed versions.. tell us more. They are getting pretty old now.. since most would be at least 5-6 years old.. and you would be pressing your luck buying them. Still if you get them for $20 or so .. I would not pay more... then heck.. it is cheap enough even if it dies in a year.
If you had asked the question before already buying one more.. then I would have said consider a mesh system like Netgear Orbi. And certainly you are going to be up for a new system in a few years.. since Apple has given up producing routers. But you need a whole system replacement.. so outlay about $400.. well you never know in a few years time where things will be at.
