Is there a specific terms to describe variant of existing genes in bacteria?
Allele

From wikipedia > allele 


  An allele is a variant form of a given gene


I somewhat disagree with this definition in the sense that it is not general enough. An allele is a variant at any locus (locus = position in the genome). You can talk about allele at non-coding sequences. Also, if you are talking about a coding sequence, you can define different sequences as belonging to different alleles whether or not their protein products differ in function or not. The proteins might even be the exact same one for two different alleles if the only difference between these alleles concerns a synonymous mutation (or a mutation in an intron).

You can feel free to define that different sequences as being different alleles by any kind of arbitrary threshold measure. You will just need to define it. For example, you can define that different sequences are different alleles only if they have at least 5% pairwise differences or if you found out by some biophysics computation that they ought to fold differently or for whatever another arbitrary measure.

The term allele is used very often in evolutionary biology and you'll find its usage in any intro course to evolutionary biology.

Rapidly evolving genes

As far as I know, there are no commonly agreed upon definition for what a rapidly evolving gene is. To me, a rapidly evolving gene is simply a coding sequence which evolved more rapidly than other reference sequences. Per se, although it would very likely involve positive selection, I don't think it is a requisite by the definition of rapidly evolving genes. It could, for example, be caused by a very high mutation rate at this particular sequence or by balancing selection causing any new rare alleles to be beneficial.