Is dark matter influenced by gravity of baryonic matter?
Dark matter attracts baryons not only by a "kind of gravity", but simply by gravity. Everything that has mass (even if it's only relativistic mass like light) attracts everything else by gravity, and thus baryons also attract dark matter.

The reason dark matter "organizes" galaxies is just that there is much more of it, roughly 7 times more, to be specific. So, on large scales, the dynamics of galaxies are dominated by dark matter. But because baryons can dissipate energy, it may collapse in clumps — such as molecular clouds — which are hence dominated by baryons.