Are you sure it's grease? The entire top of the diff on my TR4 was thickly coated with cosmolene or Waxoyl. Old and dirty, it could pass for dried up and thickened grease. It was up to 3/4" thick in some of the indentations of the diff housing. I'm sure it came from the factory that way in 1962. I found the same stuff packed all around the parking brake assembly. And, this was the first time some of those parts were fully removed from the car giving access to things to really clean them up, so I know the stuff was "original". I think they sprayed it on liberally to protect the cars from salt air during overseas shipment.
The breather is just that... a small tube that allows air to breathe in and out of the diff. Without it, you'd see leaks at the seals pretty quickly, after the first couple times things heat up. The breather protrudes a little in hopes of keeping any water or dirt from entering the diff. I've replaced it with a barbed fitting and will run a hose to a catch tank with a proper breather on it.
The cover gasket is just paper. It's easy enough to make one if you want something heavier. But it doesn't need to be very thick if the cover is in good condition (see below). I just used a coating of Hylomar on each side, not a drop of oil to be seen.
Check that the flange of the diff cover is nice and flat, and that it hasn't been overtightened and belled up at the bolt holes. If you find anything, straighten it as best possible before reinstalling. Other than that, it's pretty straightforward to install and seal up.