no no, it is not offensive. It is kind but just  informal. I don´t know how to describe...

For example lets say I am walking on the street, somebody drops her book and I give it. When she says thank you, if I answer as "görevimiz" then it sounds funny to me

görevimiz

warning: it is a fun way of saying "you are welcome" actually

don´t say it to a stranger

They are possesive suffixes.

He is wearing his sock(s).

additional info:

He is wearing the socks.

O, çorabı giyiyor. (If you mean he is doing the action now)

He is wearing socks.

O, çorap giyiyor. (If you mean he has some socks on)

It doesn´t have to be necessarily in Azerbaijani language.

This is probably from a folk song, and it is normal to use dialects and folk songs. To say "ataş" instead of "ateş" is quite common in eastern Turkey.

you are right about the literal meaning. but it is sometimes used as an answer to thank you (to your friends)

I just want to correct something small. 

When translating somthing, we should think if the direct translation is valid in the other language also

I mean, you wouldn´t say"onu seviyor" here, would you?

I think "gömleği beğendi" is more appropriate here.

Although there is no "shirt" in the second sentence (only it), I think it is better to use it in the Turkish sentence because there is no difference for he/she/it in Turkish and it sounds confusing.

Bir şey değil

Arkadaşın paraşütü istiyor mu? Gümrükte sorun yaşayabileceğini ve bir sürü para ödeyebileceğini biliyor mu?