Will non-ECC algorithms like RSA eventually become too inefficient? The strength of symmetric and asymmetric encryption schemes scales with the key length, but there is a difference between symmetric algorithms like AES and asymmetric algorithms like RSA.

For example, according to some sources, 128 bit symmetric keys are equivalent to 3072-4096 bit asymmetric keys; but to get 256-bit equivalent security, asymmetric keys need to be more than 15000 bits long.
Elliptic curve keys, however, are often stated to be comparable to symmetric encryption keys (with a constant factor of about $1/2$).

Is there an exponential relation between the key length for symmetric/ECC and asymmetric encryption schemes, and what are the theoretical and practical implications of that? Will there be a point in time where using RSA and DH/DSA will be impossible because of the huge key sizes and the resulting computations? 

How does this relate to the notion of computational security, especially the security parameter of an algorithm – is the security parameter exactly the key length, or can the key length be a (polynomial? exponential?) function of the security parameter?
RSA will never become "impractical" until it gets broken. (in polynomial-time)

As the best current algorithm to break RSA is sub-exponential (general number field sieve / GNFS). According to Moore's law computing power grows exponentially however. So simply increasing key size will always be an option.

Beyond 256-bit security EC crypto just is significantly faster than "classic" crypto. For example, for 512bits, EC crypto is 64 times faster than classic crypto (compared to private key operations using RSA).

If you are interested in key sizes you should read this paper.