Will this trimmer work on thick edgebanding?
You have to be reasonable with expectations for any given tool. This trimmer is not a router bit and therefor it will only trim veneer-thickness edgebanding.

What is veneer-thickness edgebanding?
Any edgebanding that is 1/42" thick.

Would it be better to use a trimmer that cuts both sides at the same time?
I'm not impressed with those types of trimmers because one of the two edges will often end up with tear out and and a jagged edge where the knife cuts against the grain. The knife should only cut in the direction of the grain. In the sample image below, a double sided trimmer that is moved from left to right would leave a jagged edge on the lower side of the edge banding.

Can I use this tool to trim paperbacked veneer from the face of a panel?
The trimmer is for edgebanding. A paperbacked veneer can be trimmed with a veneer saw, razor knife, or flush-cutting router bit.

Will this cut the edge perfectly flush?
No. I would not take liability for saying that a tool does anything "perfectly" but it does do a very nice job that requires little follow up work.

How much over-hang can this tool trim off?
The trimming process is completed by making multiple passes with the tool, so any amount of excess edge banding can be trimmed off. Each pass of the tool can remove up to 3/16" of material. The final finishing passes remove a very fine sliver of material.