Your pedal puts out a signal that thinks it's going to arrive in an amp input. It wants to 'see' something like 1 megaohm. If you plug it into the mixer it will 'see' 1 kiloohm = Electrical stuff! ... the D.I. (which will need a mic preamp input to plug into) will do the job of translating between the 1 megOhms/1kiloOhms thing.
Maybe the Zoom will work, if your mixer has a "line input" (this is not a mic input) as well. Then you could go "guitar-pedals-Zoom-line input on mixer".
yea, i would go USB from the zoom and use amp sim software. Currently I'm most impressed by
Using RedWirez IR's to simulate the cabinet. They really did a great job and you can get a feel for how it all works by downloading their free Marshall impulse responses at that link. Load up Space Designer or LAConvolve and go to town!
I've gotten a pretty decent rock/metal sound using the LePou LE456 in conjunction with RedWirez Marshall IR's.