Do manufactories and Customs Houses sacrifice the Strategic or luxury resources on those tiles?

I cannot figure out if using my GE or GM to build a manufactory or customs house on a tile with a strategic or luxury resource in it sacrifices that resource.

I'm also wondering if there is any benefit to building those improvements on tiles with resources.

The Customs house description says it will connect a resource to the trade network but how does that help nix the city it's worked by is already connected via trade route?

"A resource is connected to the trade network" means an appropriate building was constructed on that resource and it may now be used or traded to other civilizations. Trade routes are about connecting your cities to the capital by roads and harbors and then generating gold automatically from the connection. Trade network and trade routes don't have anything to do with each other.

A great person's improvement can connect strategic resources but not luxury resources. The type of the improvement does not matter - a manufactory, a landmark and a custom house provide the same amount of iron when built on an iron resource.

The reason great person improvements are allowed to connect strategic resources, is that strategic resource aren't revealed on the map until the cooresponding technology is researched. It would be a disaster to require someone to tear down their landmark just to get to the Aluminum that appeared underneath it.

Luxury resources are shown at the start of the game and so this grace in the rules does not apply to them.

Yes you do get strategic resources linked into your network. They do not get luxury resources.

From the 1.0.1.332 patch notes

All great person tile improvements now connect all strategic resources.

You do not however get the bonus to production or income that you would normally get from building the corresponding improvement.

The benefit of building a great person building on a resource tile is that the resource improves the output of that tile, so you can collect the tile output and the bonus from the great person's building with a single citizen. It also allows you to use Citadels to help defend important resources.