This plate does support the transmission, I think actually it supports the (rubber) transmission mount. I replaced this mount last year, and from my recollection, I placed a jack under the trans, so that when you remove the plate, the trans (and thus the whole engine/trans combination) stays in place. If you take off the plate (for which I think you also have to loosen and lower the exhaust) you may want to go ahead and replace the transmission mount, they're not that expensive and do wear out over time. It's not a difficult job, but does take some time.
I never looked at how the shifter linkage is routed so I can't help you there unfortunately.