Spongy Brakes:
Air in the brakes is common. By far the best way to bleed them is with positive pressure using a big syringe. See
http://forum.etypeuk.com/viewtopic.php ... eed#p44643 Persistence will fix this problem.
Gearbox popping out:
Gearbox popping out of gear is as you have said often trim interfering with the throw. To exclude this remove the cover in the floor that goes over the top of the gearbox and drive the car. If it is definitely not this then make sure that the linkage bushes aren't excessively worn and that the linkage has been assembled correctly. Have a read through
http://forum.etypeuk.com/viewtopic.php?f=4&t=14548. This was about a Moss box but there are pictures that you can compare to. After this I believe it is possible to adjust the shifter mechanism spring tension with the bolt that sticks out of the back of the case below and to the right of the shifter. We always get our gearboxes professionally rebuilt and I have no experience with doing this myself. This is a problem that whoever rebuilt and reinstalled the box needs to address.
Engine noise:
My first E Type after rebuild had one noisy tappet. I could hear it while driving the car. When we fixed it, the gap was 11 thou and should have been 6 thou. So poorly adjusted tappets will make noise.
Having said that I have rebuilt a number of E Type engines since. More than 10, and 3 for my own cars. I shimmed all of the heads on these cars myself. They are all quiet. So I do not agree with whoever told you that XK engines are intrinsically noisy.
Before you do anything you can listen to the noises, using a big screwdriver to your ear to locate the source of the noise. This is a surprisingly accurate method and will allow you to identify exactly where in the head the noise is coming from. It is always possible your noise isn't coning from the head at all. Water pump, alternator and the fan belt tensioner can all be culprits, as can something badly positioned hitting the fan belt or pulleys.
My approach to your car would be firstly, to tighten up the timing chain. It is surprising how much noise a loose chain can make and how "bad" that noise can be. I know someone who ended up stripping a whole engine chasing a noise that turned out to be a loose timing chain. 300 miles is exactly the point at which the chain can have bedded in enough to loosen a bit.
Then I would check all the valve clearances yourself. If they are not within one thou of correct this should be rectified.
If this doesn't fix the noise and the noise is definitely from the valve train then the next potential culprit would be that the tappet buckets are loose in their bores. An XK head rebuild must involve re-cutting the bucket bores and replacing the buckets. We routinely use oversize (1.4") long skirt buckets with the clearance cut to exactly 0.5 thou. We use buckets from XKs Unlimited because our machinist says they are very consistent in diameter.
Other possibilities after that include looseness of the tappet guides in the head. We routinely fit hold-down kits to both inlet and exhaust sides. If this hasn't been done to your head it should be.
After that, you are left with the camshaft (damaged lobes, bearing problems) and oil starvation to the valve train.
Lobes can be inspected for witness marks suggesting wear or damage etc. Cam bearings rarely cause trouble in my experience but it is possible to bend a camshaft if you're a gorilla so that would be worth checking for completeness. The oil supply to the valve train comes via the pipe at the back of the engine. If you reuse your old one it needs to be thoroughly cleaned. If someone has fitted a braided line get rid of it and refit the original. Some braided lines we have used are made up from brake fittings and don't provide adequate flow which will make the whole valve train noisy.
Bottom line:
We restore cars and charge people a lot of money for doing it. If the customer isn't happy with something we fix it. So should your restorer.
Handing a customer over a car with bad brakes is dangerous and reflects poorly on your restorer.
The gearbox issue is annoying but should be a relatively easy fix.
XK engines are NOT intrinsically noisy if they are built correctly by someone who knows how to do it properly.