Bowtie2 should work as well as Bowtie1, since bowtie2 should not return a gapped alignments if a contiguous one exists. Generally speaking. However, some people prefer Bowtie1 since you can guarantee "--best --strata", which Bowtie2 does not do.
There are better aligners for miRNA data, though, which also include analysis of the alignment context to give a more accurate picture of expression: see MirDeep2 and ShortStack (GitHub link).
HTSeq-count will quantitate multireads if you force it to. It depends on two things: one, mapping quality of multireads (which is usually very low or zero), and two, any SAM tags indicating a multiread (which is only the NH tag, according to the documentation).
HOWEVER: only Tophat2 uses the NH tag -- Bowtie1/2 do not -- Bowtie2 uses the XS tag; not sure about Bowtie1.
But in any case it is only a SAM tag, which can be stripped. For instance, to blind htseq-count to multireads from a Tophat bam file, set "-a 0" to stop filtering on mapping quality, and strip the NH tag in-line:
samtools view in.bam | perl -pe 's/\s+NH:i:\d+//' | htseq-count -s no -a 0 -m intersection-nonempty - genes.gtf > counts.txt
If htseq-count ever uses other tags, then just strip those too.