Question about discrete logarithm problem DSA and ECDSA both are based on discrete logarithm problem (and elliptic curve based DLP)

And ECDSA protocols resembles DSA.

My question is,

Can every protocol based on DLP be changed (substituted) to a protocol that is based on ECDLP?
To convert every protocol based on DPL to ECDPL based protocol there is a limitation as EC uses generators from a very large prime(or binary) fields. For an instance consider ECDSA which is an EC analogy of DSA, in ECDSA there is a Generator G (used in calculating public key) is from a large prime subgroup of EC where as in DSA the private key (x) is just a random number such that  0 &lt; x &lt; q where q is N-bit prime number.