Does one Kohen have a mitzvah of vekidashto to another Kohen?
Answer:
In general, the mitzvah of vekidashto applies to a Yisrael with respect to a Kohen, due to the greater kedushah of the kohen. It would not apply to a Kohen with respect to another Kohen.
This is certainly true concerning the mitzvah of preceding a Kohen to a Yisrael: concerning one Kohen to another, no concept of precedence can apply.
However, it would seem that even concerning the prohibition of making use of a Kohen (see Rema, Orach Chaim 128:45), or of degrading a Kohen (see Chafetz Chaim, Asin 9), the prohibitions do not apply between Kohanim themselves, who are on an equal level of kedushah. See Biur Halachah 128, s.v. assur, who writes as a possibility that the prohibition of using a Kohen does not apply to another Kohen, and quotes this stance from a number of authorities.
According to Taz, however, it appears that the mitzvah would apply, for the Taz applies the mitzvah of vekidashto even for a Kohen with regard to himself–and the more so with regard to another Kohen.
