What is a diff - 2007 Ford Focus (3 Door) Hatchback

80-90 oep diff oil unless it is a limited slip diff in which case you use special limited slip diff diff oil
don't know if it is limited slip diff diff
you can check by jacking up the rear end and having a friend turn a rear wheel
by holding one wheel the other wheel will be turning in the forward direct as that is what limited slip diffs do diff do ( both drive )
other than that call your local manufacturer service center and ask

manual transmission --- 80w-90 for the gearbox and transfer case
diff is 80-90 oep if a straight diff and different if a limited slip diff gear box is around the 2.5 ltrs as is the diff each
LiSiD oil can be used in LiSiD diffs and straight diffs but straight oil cannot be used in lisid diffs
the transfer case uses the same as the box but the quantity you will have to check in the lubrication section of the user manual
many user manuals recommend a lighter straight oil for transfer cases and boxes ( 30 straight oil) but I find that the light oil promotes bearing wear and gear problems as the oil squeezes out under load

suggest that you talk with a diff reconditioning specialist center
if you have excessive pinion to ring gear clearance the diff will be making terrible noises not a clunk
what you have is worn side gear washers in the diff carrier and that is not essential to repair unless you get diff noise on turning corners
All diffs have play in the diffs that most people say is pinion /ring gear problems when it is not
the clunk noise comes from engaging a gear and the diff housing reacts to the load ( radius rod bushes , stabilizer bar bushes etc), the side gears take up the clearance, and the
"U " joints or cv joints take any wear
Best advice is--- if all you have is a clunk ( check everything outside the diff first) and the diff runs quiet under acceleration or deceleration, DON'T fix what ain't broken

You don't say what type of car you have and what mileage it has run so far.

Most likely the whine is from a very worn pinion gear.

Other than replacing or rebuilding the diff there is nothing you can do other than drain the old diff oil and replace it with new oil. ONLY use the manufacturer specified differential oil in the correct grade. This may quieten the noise a little but any difference will be minimal. It may help extend the life of the worn diff a little and it will get rid of any debris suspended in the old diff oil..

The diff drain plugs are often magnetic to pick up metal to prevent it being recirculated so check if there is much in the way of metal filings on the drain plug when it is removed.

Without disassembling the diff no one can tell you how long it may last but they are expensive to replace so I would be replacing the vehicle sooner rather than later if you don't want to replace the diff.

Until you sell the car, drive it gently when moving off from a stop and when accelerating to minimise stress on the diff.

If you are going to replace the diff oil yourself the car will need to be level and ensure you loosen and remove the diff oil filler plug before you remove the oil drain plug. If you drain the oil and the filler plug is stuck you will be in trouble. It will be a lot easier to just buy the correct diff oil and take the car to a reliable mechanic to drain and refill the diff for you while the car is up on a lift/hoist.

Yes it is a wind up of the axles and not the drive shafts. Your car has a limited slip diff and it is locking up.
When you are accelerating the diff locks up and your wheels are rotating at different rates while turning. This causes the axles to twist until the pressure causes the Limited Slip Diff to slip.
My guess would be that someone has changed the diff oil with non Limited Slip Diff oil.

remove the axles, undo the back plates from the housing. remove the diff carrier bolts . take care to keep shims to the same side of the bearings. pull the diff centre out. remove crown wheel from the diff carrier . fir crown wheel to the new diff carrier (limited slip centre) . Refit carrier into diff housing. Best advice --have an accredited diff shop do the job for trouble free operation.

diff overheating is from incorrect mesh of the crownwheel and pinion
that can be from too little clearance , too much clearance of failing bearings and too much preload on the pinion bearings
it can also be from incorrect oil level in the diff to the bottom of the threads of the fill plug
it is time to pull the diff and check the bearings
the problem will most likely be the bearing behind the pinion as that one takes all the load
diffs normally run hot but it is the escaping oil that is the problem
that could be from bearings on the pinion allowing the lip of the flange seal to not seal properly

Take of the right or left side wheel and you will see a small dipstick for the diff

If I recall correctly, the axles have splines into the diff and also into the hubs. Removing the diff requires the trailing arms to be disconnected from the control arms to swing the axles enough to slip out of the diff.

1 out often does it but yes, you have to pull one clear of the pumpkin by about 2 inches. Sometimes easy as pie sometimes drift hammer.