I am forging 3 mantel brackets out of mild steel for a 99"x 9" x 4 1/2" old barn timber that weighs 90 lbs. What type of hardware would you recommend to secure it to the studs? Each bracket will have 2 holes on the vertical run to attach to the wall/stud and will be 1/4" thick. I can get #20 (5/16 shank) pyramid head wood screws that are 2 1/2" long. Or, should I go with a 3/8" to 1/2" lag bolt? What would be an ideal length for these? 
Walls – Hardware recommendations for heavy beam mounted to wall
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You do not want to cover over the window with a TV. First of all there is the glare point that Paulster2 points out. That alone should make you take stock of this situation and see what a bad idea this is. However there are a number of additional considerations as well.
Windows have a role of safety exit possibility if you ever get caught trapped in a room during a fire. This for both outbound escape and for inbound access by emergency personnel.
There is a reason windows are added to a house. Obviously it is to allow light in so that the room does not look like a cave.
If you have no usable wall space to mount the TV then get your self a console table. These do not have to be that expensive and you can even get them with an integrated post assembly that permits mounting the TV bracket with tilt and swivel capability. Best with this approach is you can move it around if your change your TV room layout or go off to a different house.
Talking about moving. Say you did move after you had mounted a TV over a window. Think of the mess in the window trim that would need repairing before you could sell.
Lastly, putting a TV over a window is just weird.
I am certain that if you mount that sink to the wall using only the two holes in the tabs, no matter how strong the mounting screws nor how hard the backing, the cast iron tabs will snap off.
In the worst case, the tabs will hold the sink in place for days or even weeks, then break suddenly when you are barefoot in your underwear and the sink is full of soapy water and a wet cat.
Don't even think about using this exquisite antique until you have figured out how to construct supports between the front mounts and the wall.
Other than the thickness of the backing board, you have a lot of freedom to determine dimensions. The backing board should extend downward as far as conveniently possible to spread the levering force. Using the backing board also allows you to position the sink without regard to the wall studs.
The screws and bolts that pass through the sink should be as large as will fit thru the holes, not only for strength but to avoid lateral motion.
Here is a rough sketch of a possible installation:
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Best Answer
I think your choice of 5/16" size lag screws, 2-1/2" long, is fine. I would say 3/8" is overkill but if you just like the look of the bigger fastener, it would be fine.
I'd actually use something a little smaller, but longer - 1/4" size, and 3" long. (And washers, whatever size you use, I'd use washers.) One of those is adequate to support 90#. I'd also drill a pilot hole, 3/16" bit works well for 1/4" fasteners in most wood.