You can certainly use a "glass" 10 gallon tank as a refugium - many people do.
As Heavenly Damsel said, you'll need another overflow on the fuge to get the water over the edge of the glass tank, and back to the sump. Or have the sides of the 10 gallon drilled ( not recommended ! )
Another method is to use a plastic or acrylic container for the refugium, and cut holes in the side of it, up near the top, facing the sump.
Install bulkhead fitting(s) in these holes.
These holes/fittings MUST be higher than the top of the sump ( wet/dry ) sides in order for this to work.
Rubbermaid type containers normally allow you to use a larger size than many glass tanks.
Using some cheap pvc elbows and short lengths of pipe, you can easily make one or two "spillways" that allow the water in the fuge to "flow" through the bulkheads and "fall" into the sump ( wet/dry ).
This eliminates the need for the additional U tube overflow on the refugium. I don't like to use U tube overflows on LOW FLOW situations. It's possible they will accumulate air in the upper curve of the U, and lose prime/siphon.
As far as feeding the water to the refugium, yes you can certainly rob some of the draining water from the display tank as it falls to the sump. Teeing off the drain line will allow this.
Personally, I prefer to rob some of the return pump water with a tee in the return line, and redirect this water to the refugium.
What goes up must come down.
I will adjust what goes up - but will not adjust what comes down.
I let it come down unrestricted.
Additionally, the refugium feedwater line now is under pressure, not relying on gravity. This makes it a lot easier to run your tubing/piping. You force the water over to the refugium, can regulate it very accurately with a small 1/4 turn ball valve ....... and not disrupt your tank draining.
I avoid wet floors as we all do - this set up has worked 100% since February/March - and didn't cost much at all. Everything excluding the Rubbermaid containers is available at Home Depot.
If you aren't into using Rubbermaid, then glass or acrylic is still a good option.