Using POE To Extend Wireless Network

My b/g/n router is at one end of the house and on the second floor needs to be. I currently have a b/g/n wireless bridge hooked to a LAN port of an b/g/n router that I am using as an access point almost in the center of the house, about thirty feet away with one wall between them. I get good connection speed on an Internet TV that's about 7 feet below the access point with one floor/ceiling between. I get a reliable but slow connection to an Internet TV that is about 20 feet from the access point with two walls between. I get intermittent, slow connections with portable devices near the opposite end of the house on the first floor. Measured horizontally, they are within 15 feet of the upstairs TV that has a slow connection and about 9 feet below it.

The ceiling of my unfinished basement stairwell is a few feet below my router. I am thinking of putting a wireless bridge in the stairwell and connecting it with ethernet cable to an access point located in the crawl space, which is very dry, under the house directly below the upstairs Internet TV. It would be about 12 feet below the TV with a floor and a ceiling/floor between. That location would put it within 15 feet of the wireless devices on the far end of the house that now have a poor connection, with only a floor between. My primary goal is to get a better connection for them, but I'm hoping for a better one to the upstairs TV. My house is of frame construction with mostly plaster interior walls.

I want to use a POE access point under the house because I think it would be easier and safer than running a long extension cord. Looking at POE devices in catalogs, it is often not clear to me whether the device in question expects to be powered over ethernet or expects to power other devices over ethernet.

I have the following questions:

Does my plan sound feasible, and would it be expected to accomplish what I want?

How can I be sure that, when I order a POE access point, it expects to receive power over ethernet, not provide it?

Can POE devices be cascaded? If I do this and find that I did not get the coverage I hoped and, therefore, want to add another POE AP, can I add a POE hub or switch and run cable from it to the additional POE AP? If so, is there any particular monclature for devices that expect to both receive and provide POE? Can such devices receive and provide over any port, or do they have to receive power on a designated one and provide it to the others?