Question: If I were to publicly display the 6 digit code from my google authenticator, would my account security be diminished beyond it's valid life time?

What if it were the code and the time it was generated?

What if it were N sequential codes?

I'd like to know why or why not for each case.

Answer: 
No. Thats the whole point of TOTP/HOTP. That it should be infeasible, given any sequence of valid codes, to generate future codes or recreate the seed.

The reason is that the TOTP/HOTP algoritm, use HMAC-SHA1 as hash, given the "challenge" (either current UNIX time or a counter of usages) and the secret "seed" as key. The resulting hash output is then truncated according to a very specific rule.

This would mean, in your first case, it would be impossible to do, as multiple valid keys will come up with the "correct" 6-digit code but will turn out to be incorrect as future codes from these "valid keys" will be invalid.

To successfully mount a bruteforce attack against a seed, you would atleast need 2 valid auth codes, possible more, and even in that case, its infeasible (eg, practically impossible) to do it.
The first OTP definition in RFC 2289 based on S/Key had even this design requirement.

It was not ment to implement two factor authentication but to merely protect the password or passphrase. It was simply used to avoid the password crossing the network - obviously it was assumed that the transmission could be evesdropped or unencrypted.

So as already stated, it is a fundamental design in all OTP systems (S/Key, mOTP, OATH) to not allow the guessing of future passwords based on used passwords.

**But:** With an event based algorithm like HOTP you can of course create OTP lists. If you generate an OTP value *now* and post it on the internet but don't use it for login, everyone else can use it. It only gets invalidated, when this very value gets used or a later value gets used.