Birth certificate? SSN?

No, you don't need copies of your BC or SS card. All you need is your passport, and your drivers license if you plan to drive. I don't know where you are going, but if you do plan to drive in a non-English speaking country an International Drivers Permit (IDP) is often required.

I like to use my DL as collateral when renting headphones, etc. I don't want to take a chance that my passport will go missing. An expired DL is even better.

Nope. You already have exactly what you need.

You may be asked for photo ID if you use a credit card (not likely) or as a deposit if you rent an audio guide at a sight (very likely). Use your driver's license, not your passport. If your DL gets lost, you can easily replace it when you get home.

For credit cards and ATM cards, make a note of the emergency phone numbers on the back. Also notify the issuers what your travel dates are and what countries you are going to.

All you legally need to cross a border is your passport, and if you are intending to drive a car, your Driving Licence and possibly an IDP (International driving permit).
Birth certificate, Social Security card etc. are not internationally recognised documents and have no validity outside your home country.

You mentioned your health insurance card. Does your health insurance cover you in foreign countries?
Some do and some don't, if you don't know you'll probably want to check.

Note re Social Security cards, from a former SSA specialist. This should never be carried anywhere. The only place that could need to see your actual card is an employer of a new hire who is suspicious that the name/SSN combination is not legitimate. Carrying the card anywhere risks its loss, and the possibility of someone else using the name/SSN combination, including to reference you. There are many ways to search the net on someone and dig out an actual DOB and parents' names. The last bit of the puzzle is finding the SSN that goes with that name, and a lost card gives that up.

In spite of the advice to never carry your SS card, if you are chronologically challenged and on Medicare you will need the card to get any kind of medical care in the US. Since you never know when you might need medical care in an emergency, that implies always needing to carry it.

I'm on Medicare and I've never been asked to show my SS card when seeking medical care in the U.S. or abroad. Maybe plans differ?

If you are on Medicare and go for medical assistance, they do not need to see your card, they only need your number. If your name and SSN do not match up when they submit the bill for payment, it will be bounced back to them immediately and your provider will be back at you just as fast. Losing a Medicare card puts you open to identity theft even faster than losing the SSA card, as in most cases your DOB is now known within the confines of a month (the hospital insurance starting date is almost certainly going to be the month of age 65).

If a US provider is having an issue with you over your claim of the Medicare #, which they should not, you can always ask them to call the SSA toll-free number, they will verify this.

Question for the Original Poster:
You mention carrying a health card.
Does your existing health insurance cover you in foreign countries?
Some health insurance covers you in Europe, and many don't--if you don't know you'll probably want to check.

You don't need to take your SSN card with you, I think lots of folks have given you plenty of advice on that, so "please" heed their warnings. I do leave a copy of everything with our daughter though, so she has all the information on all of our medical, SSN numbers, total travel, and flight plans. We are both on Medicare, so she copies of those plans, and the appropriate information regarding our care plans, since we carry additional insurance, besides Medicare.
You won't need a copy of your birth certificate either.