What is a word and how much is a word allowed to change to be still regarded as the same?

E.g. there are many reconstructed Proto Indo-European roots that we still use in words today - however, they are just word roots not words (reconstructing the actual words is more controversial), they have undergone phonetic shifts on their long way into English and they are reconstructed (and not actually witnessed in any inscriptions etc) at that.
Our earliest inscriptions of Indo-European languages fall in the the 2nd millenium BCE but reconstructing Proto Indo-European roots aims at the time when the Indo-European peoples still lived in their Urheimat, which is usually assumed to be at some point in the 7th to 4th millenium BCE.
So with some creativity you could argue that words like "father" (*ph₂tḗr) or "work" ( *werǵ-) might be 5,500-8,000 years old.

If you want to base your choice on actual inscriptions instead of conjecture I'd have a look at Akkadian. It seems quite possible that you'll find Akkadian words which are also present in Hebrew and that have entered English from there. That might get you into the mid 3rd millenium BC if you are lucky.

Do you mean in English, or in any language? Does it have to be spelled the same way? Pronounced the same way?

What counts as "still using it?" Does, for example, latin count, even though nobody speaks it as a first language?

While you might get a good answer here (we have several linguists among our flaired users) and I don't want to discourage anyone from asking or answering I think you'd have an even better chance if you asked at /r/asklinguistics.