Is there a 3-D alternative for elliptic curves?
So, i've just started to figure out how encryption works. 


From what you've said, I fear that's not true. 


  As i understand the most widely used method to exchange encrypted data over untrusted network is Diffie-Hellman protocol. 


Partially true, as DH is used as key agreement protocol to further derive session encryption &amp; MAC key. DH is asymmetric, where as encryption and MAC (message authentication code) are symmetric. 


  So, original idea was to use big prime numbers (1 number, 1-D) ... After a while elliptic curves (x and y coordinates, 2-D) was discovered, so now everyone using them. 


The original realization of the idea was to use a big prime number, but those numbers that can be represented in binary system have a exploitable structure make possible attacks faster than brutal-force search. 

DH is still realized using scalars, except this time, the scalars are points on elliptic curves over finite fields. 


  Question is - do elliptic curves have more cryptographic strength than classical approach because of 2 dimensions?


No, they are stronger because they don't have the same exploitable structure as (big) integers. 


  Is there researches related to 3-rd (or even more) order polynomes that potentially can provide even more cryptographic strength than elliptic curves?


Yes, one of my questions here attracted this comment, but I haven't thoroughly comprehended it, I advise you read it with care if you have time.