When is sağol used and when would you use teşekkür or teşekkürler. Are they actually interchangable and, perhaps more importantly, is there anywhere that it would be inappropriate to use any of them?

"sağol" literally means "be right"
and i believe that it is a more informal way of saying thanks.

Both sağol and teşekkür ederim (or teşekkürler) means "thank you".

In the army you must use sağol and you must not use teşekkür ederim.
When the commander comes and ask the question:
-How are you soldier?
Everybody must shout :
-Sağol.

"Sağol" is more mannish and I wouldn't say sağol to a lady. It is OK for men to use sağol towards closer male friends. Sağol is informal. I think a sophisticated or polite female also wouldn't use sağol at all.

You can say:
"sağol dostum" > thanks buddy

If the prime minister would visit the farmers they would say "sağolun" which is the plural form but can be also used as polite singual. So with "sağolun" we have the mannish and strong and polite way of saying thank you.

(Sen) sağol.
(Siz) sağolun.

Teşekkür ederim (or teşekkürler) is more polite and formal. Also "Çok teşekkür ederim" is very common.

If my car had stoped on the street and a few men had come to push it I would say "çok sağolun" as "teşekkür ederim" would look too girlish to me in that situation.

Is there anytime where I could get it wrong to the extent to accidentally insult someone by using the wrong term? Bear in mind that I don't intend to join the Turkish army

No bod it will be alright for you in any case. Actually the difference is not so clear in dail life.

I changed my mind about female using sağol. I realised that female use the polite form of it, "sağolun", commonly.

I was told to use it as "no, thank you", when offered, for example, more food at the table. I was also told that the appropriate accompanying gesture was the open palm against the chest. Was I wrong in doing so?

Sağ ol is used in most cases where somebody goes out of their way to do something for you, or do something for you that isn't in their job description etc.

I wouldn't say it's a particularly manly thing to say. It can be used for both men and women.

It's better to say sağ ol to someone than teşekkür ederim if they have gone out of their way to help you, it's just more polite.