It is worth a try cleaning it... You probably could get the work done locally. There should be plenty of camera repair techs in the Pacific NW of the US experienced at cleaning fungus! Might avoid the cost and hassles of shipping, and get the work done at a lower cost locally. Check around.
It depends upon the type of fungus and how long it's been there, whether there will be any residual damag from the fungus (which, when it occurs, is usually etching of the glass or staining of any coatings on the glass). But you really can't know until you try.
I've cleaned a few lenses infected with fungus over the years and had about 50% success... but most of them were old vintage lenses and might have had fungus in there for a long, long time.
If the fungus is at all centered on one or more elements and any size at all, it probably is effecting image quality, just that you don't know it because you have nothing to compare. The 300/2.8s are so sharp to begin with, that a bit of loss of IQ might not be that obvious, until it's closely compared alongside images form a lens without fungus.
Yes, fungus spreads by spores and it can get into cameras and lenses stored near or used with an infected lens. It would be more likely with a zoom that moves a lot of air in and out of the lens, as it extends and contracts, but it might happen with any lens. After all, it managed to find its way inside a pretty well weather sealed lens in the 300/2.8. Just to be safe, I'd keep the lens stored separately... perhaps sealed in a trannsparent plastic bag with some dessicant to reduce moisture around it and discourage any further growth of fungus.
Also, most fungus thrive in darkness, while UV light and daylight kill many types. so until you get it fixed, you might expose it to sunlight, leaving the lens caps off. If you seal it in a transparent plastic bag that would be possible while keeping it sealed up to protect everything else.