1. How good is a hydrotest certification good for? If the tests are not recently certified, how much am I likely looking at to get them recertified?
It depends on the DOT rating of the cylinder. DOT 3A cylinders are required to be tested at 5 year intervals. DOT 3AA cylinders can be tested at 5 or 10 year intervals, depending on the marking accompanying the most recent test date. DOT3A and 3AA cylinder types are used for oxygen and other high-pressure gases.
DOT 8 (acetylene) cylinders only need to be retested if they're damaged, dented or very rusty, and if so, they're sometimes removed from service instead of retested, depending on the amount of damage.
Look for the most recent date stamp on the neck or shoulder of the oxygen cylinder. If it's a DOT 3AA rated cylinder and the most recent test date has a star symbol after the date stamp, the hydrotest is good for 10 years. Otherwise it's only good for 5 years.
In addition, some cylinders may be marked with a + sign after the hydrotest date stamp. That means the cylinder can be filled to 10% higher pressure than the DOT pressure rating. Just mentioning that in case you see that mark and wonder what it means.
When needed, a hydrotest will cost you around $20, +/- $5. Price varies a little by location and by the company doing the testing.
If you rotate your cylinders out frequently, you may never get caught with a hydrotest charge. It often only becomes an issue with DIYers who don't do much welding and the cylinders sit for years and years between refills.
2. Are ownership documents required for all tanks or only those with a gas supplier cast into them (Airgas etc.) The actual document, is it basically just the original bill of sale?
That varies significantly by location and by gas supplier. It also sometimes varies by the size of the cylinder. Most of the smaller size cylinders are considered customer-owned cylinders. Larger cylinders like the standard industrial sizes are more often questioned.
As Jere said, different places have different policies, so you really need to ask whoever you choose to use as a supplier.
In places where an ownership document is required, it usually has to be an original bill of sale from a welding gas company, not a scrap of paper from Joe Blow at a garage sale or Craigslist, unless it's also accompanied by the original bill of sale in his name from a welding gas company (which isn't likely to happen at a garage sale or Craigslist).
Once you've got an account with a gas supplier, you shouldn't need to prove ownership with each refill, but if you change companies, you may need to provide proof of ownership again to the new company.
In places where sales receipts aren't generally required, it comes down to what is actually stamped on the cylinder or marked on the neck ring, which is why it's generally best to buy at least your first set of cylinders from a welding gas company. At the very least, you should be cautious about cylinders that have a gas company's name stamped on them or embossed on the neck ring, unless they're accompanied by an original bill of sale.
FWIW, in over 40 years of owning many different types and sizes of welding cylinders, I've never been asked for proof of ownership by any of the many companies I've done business with, but I also worked in the welding gas industry at one time and know exactly what to look for on cylinders I've bought used from private parties. Once again, not every welding gas supplier and geographic location is the same, so YMMV.