That plate is pushing it. I'd weld it on the back side all the way around at the very least. As far as the rubber mat thing you have between your core support, and your bumper, that can come out, no problem, the only thing its good for is catching mud, but it does help deflect some mud from the front of the radiator. The air dam somebody else was probably talking about is the black plastic chin piece they put on some bumpers that hangs down a few inches like this
http://z.about.com/d/4wheeldrive/1/0/P/W/1/Shadowhunter_GA_96F150_RR.jpg
The reality is, you should have a removable winch mount, similar to this:
http://www.jeep4x4center.com/jeep-winch/mile-marker/images/mile-marker-cradlemount-banner.jpg
Harbor Freight sells one (they sell Chicago winches too, which I have on good athority that the 12,500lb is a good one) or you can make one for CHEAP.
You WANT a removable for several reasons, the least of which is to move it to the front or rear, as required per how and where you are stuck.
You can use these:
http://image.tradevv.com/2008/08/11...sc175-bl-ups-connector-plug-charging-plug.jpg and run power cables to the front and rear, with that plug, and they have male and female plugs, so its as simple as putting your winch where you want it, and hooking up the wires.
The reality is, you don't need a winch good for 1.5x your vehicle weight in your case, because you are not pulling your truck over rock walls, or super deep mud. A 9K winch will pull a 5k truck out of thick sucking clay mud up to the axle tubes with no problem. You are wheeling alone, and need to make sure you don't get stuck and have to rely on someone elses good graces to get you out.
You never said, what is this vehicle setup? Give us Cab and Bed style, engine, transmission, tire and wheel size, how many fuel tanks, if it has a tool box or other weight adders that are always present, etc. This is what you should start every post you are asking a question with
Example: I have a Regular Cab, long bed 1988 F-350 4x4 5 speed with a stroke'd EFI460, 38x12.50" TSL tires on 16.5x9.75" wheels, stock axles, and lockers front and rear. It has a tool box with a couple hundred pounds of tools and recovery gear, custom fab'd 1/4 plate bumpers, and dual fuel tanks.
This gives everyone an idea of what you are working with, an idea of how much it weighs or how capable it is, so we can help you further.