I would not be overly worried about that quantity of oil there - just keep topping it up. I put about 300ml into my t-case every 3 months based on about 1000miles a month. I will change it when I have a free summer afternoon and the list of things to do is MUCH shorter!
You could take the plate off and drive flange and renew the oil seal if you wanted to. You would need to put it in diff lock to use the engine/vehicle to stop the flange from turning, but then the whole LR will start to drive over you unless you chock it, or have someone stamp on the brake. Or you can get a piece of steel and drill holes in it that line up with two of the prop bolts with a space in the middle to get access to the nut and use that to counter hold the flange while you undo the nut.
The oil catcher as you can see directs the oil down the back of the plate, away from the drum, so unless you can see oil contaminating the drum I would not worry about that - and would not suspect it would be causing the brake to slip.
Your shoes look like they have plenty of meat left on them; if the handbrake is slipping I would simply adjust it.
On the opposite side from the expander is the adjuster, if you look where the shoes sit into the expander and adjuster (top and bottom) you will note there is a little piston with a channel in it, on the adjuster you can wind the pistons out slightly so that the shoes sit slightly closer to the drum, on pulling on the handbrake it will bite sooner than it did before. SORTED!
So I would check the adjuster is moving OK, if not now is the time to strip it off, clean it up and rebuild it (very simple, it has a threaded section which winds in and out and acts on a diagonally cut piece sitting on rollers which will move the pistons in or out. If it's seized up, pull it apart, clean it and then pack it with grease and refit, the rollers can sometimes rust up and you will need new ones, but if your lucky they will be OK and you can get the adjuster working. If they were totally stuffed as a temp fix a piece of steel rod cut to size would let you rebuild it and get it working in the mean time.
Put it all back together and wind the shoes out a little bit then try your handbrake.
What happens is the crud can build up inside them stopping you from pulling the brake on hard, because the shoes can't press out onto the drum with enough force but once it rotates the crud just crunches away and your hand brake will slip. Every now and again you can clean the handbrake up a bit by pulling it on slightly while the vehicle is moving at speed with the ratchet button pushed in so you can just apply the shoes to the drum slightly and clean up all the crud. Applying the handbrake when the vehicle is in motion is highly unadvised; you just want to give them a quick blast.
Hope this helps.
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