I would love to know if this can be used as in English when talking about people, like in: "These guys are old school."

Note the correct spelling IRL would be "Old skool".

Don't say "Old School" because you will immediately lose any credibility of being Old Skool

After listening a couple of times on this sound clip, I can't help hearing старие and not старая школа. Both fast and slow versions give this impression. Am I the only one hearing this?

Старый школа is what I hear.

thats the way we go

Can it also have the same meaning as in English "my old school" -- the school I went to 9even if the school is physically new)?

I have seen DL make a big deal out of whether something is a sentence or not. Allow me to point out that this phrase does not start with a capital letter and does not end with a period. Therefore, it is not a sentence as described.