Can NORX32-f be used in a Salsa20 style stream cipher?
NORX's $F^l$ is meant to behave as a random permutation (modulo some known harmless properties), given enough rounds. $F^l$ is similar to the ChaCha double-round in all but the addition operation, which is replaced by $H$, and the rotation constants.

NORX should be usable in a Salsa20/ChaCha-style mode; a break in that mode would likely spell trouble for the NORX sponge mode as well. Note that the last step, adding in the initial state, can be simply done with xor; no need to use $H$ in there. Since you lose authentication by using this mode, the cost of an independent MAC should also be taken into account.

Another option to improve NORX32's performance would be to use the parallel mode. With, say, $8$ lanes you could use AVX2 to compute all of them simultaneously. Since NORX suffers from being a purely sequential mode, this would improve performance by a factor of 2 or more. This would not, however, be an 'official' CAESAR-submitted mode.

That being said, I cannot recommend using the NORX permutation in these modes, nor can I recommend using NORX in any form whatsoever just yet. This is a cryptographic primitive in its infancy, and its strong security should not be assumed.