Your air filter housing should look like the one pictured.
Note that I have replaced the oil bath with a modern paper filter. The pictured air filter element is a perfect fit from O'Reiley.

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Your fuel mileage is fabulous for one of these cars!

Every time I rebuild a carburetor, I think "Why did it take me so long to do this?". They just run so much better, even when nothing is obviously wrong.

I would agree with other posters here that the lifter noise may go away with a change of oil and it won't hurt anything to run the engine with the tick. The head does need to come off to replace the lifters but Packard heads are easy to remove. All the nuts are easily accessible and there is very little else in the way. Does your 359 have it's original aluminum head? The aluminum head will have "359" cast into it. If it does, then you definitely want to remove it and inspect the mating surfaces. Your head gasket problem may be related to a corroded head or an improper head gasket.

Many of these engines have a replacement cast iron 327 head but your car came with a 359 ci engine with an aluminum head.

If you eventually do need a new lifter, i can recommend the Kanter lifters. I put in a whole new set in a 1954 359 and it is the quietest engine that I have ever owned.

These are the motor mount bushings you need. They are sold in pairs that go together back to back in the motor mount itself.
They are fairly simple to install and will correct your fan belt clearance problem.
Thanks to Max Merritt for the picture.

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Some pertinent questions that nobody has thought to ask:
Do you have any experience working on cars?
If you have to pay for any work that needs to be done, then that changes the calculation considerably. Even minor repairs can add up to a big bill.
If you are a competent mechanic, these cars are simple and mostly easy to work on. The parts are available and not unreasonably priced.

Have you ever done any body work or painting?
I happen to enjoy painting but most hobbyists don't feel comfortable taking this project on.
If the current paint is serviceable, maybe buff it out and live with it for awhile even if you don't like the color all that much. A good painter could match the paint and do some touch-up repairs without re-doing the whole car.

Every person who has commented on this car is looking at it based on their own capabilities. Everything looks better in pictures so only those who have seen the car in person know what it really is like.

A couple of screenshots from "Marlowe"
Philip Marlowe (Liam Neeson) drives a 39 120 Coupe

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I think that the compression seems good for this engine. But if you still want to put in new rings, I have a set of Standard rings and a set of .02 oversize rings that I have no use for. These are new in their somewwhat beat up boxes from Packard. I would make you a good deal on these, so if you need them PM me.

Another thing to check, is the condensor body grounded? I once had one work its way loose and it would lose its ground at random intervals. Took me a while to find it.

I see you have already found Mike's Carburetor on youtube. He has an excellent video about correctly setting the metering rods. www.youtube.com/watch?v=4QgjJYjErZ8
It took me less than 15 minutes with a screwdriver to solve a couple issues with mine. I was running too rich.

Re: Transmission additive. It was a transmission specific product and I'm pretty sure it was by CD-2 but it has been a few years and I may not remember correctly. Maybe it was a Lucas product.