Re: Code 303.
I don't know of any way to tell for sure (valve(s) versus head gasket) without pulling the head. If you remove the head, you're going to need to replace the head gasket. So it seems to me you're looking at a new head gasket one way or the other (i.e., whether or not head gasket leakage is causing the compression loss).
This means the only remaining question is whether or not you've got a valve problem (sticking valve; burnt valve; whatever ...). You really can't tell without pulling the head.
If you're going to have to pull the head, you might as well invest in a valve job (regardless of how much valve damage / wear there may be). My own experiences (a long time ago) led me to make a rule that it's wise to do a valve job whenever the head is removed. Nothing is as maddening as having to pull a head off a second time 'cause you were trying to be too tricky the first time.
Yes - one would think a blown head gasket should result in loss of oil and / or coolant. But it's possible for the 'seal' (between head and block) to be leaky for "gases" without much fluid transfer.
It's still possible you have ring failure / damage in #3 that's severe enough to prevent the added oil from giving you temporarily better compression in that cylinder.
At 368,000 miles, I wouldn't rely on a piecemeal approach (fixing one little part at a time) - I'd go for at least valves and rings all at once on general principles. I've seen too many cases where being too optimistic and trying to get by with fixing just one thing ended up costing more than doing it all the first time.
Just my .02 ...
Yes - the head can be reliably rebuilt provided it's not structurally damaged (e.g., cracked, badly gouged).
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