The spidery little lines or mat-like formations aren't the actual fungus anyway, they're what it leaves behind.
I am not remotely worried about transferring spores between lenses, as others have said the spores are everywhere and short of only using your gear in a clean room you won't avoid them. Fungus is not contagious - people who say that one lens "infected" their other lenses are like the people who claimed that TV ruined your eyesight. It's more a case of people noticing that they needed glasses once they had a regular need to see something clearly from more than a metre or so, or in this case if you store all your lenses in less-than-perfect conditions then you will get fungus on all of them eventually. Lenses I have cleaned up and dry stored show no signs of re-infection two years after cleaning.
Those old leather or leatherette lens cases have a lot to answer for IMO, the three lenses I have which had presumably lived in them from new all suffered from fungus. If you get your lenses slightly damp (say shooting on a wet day) and then stuff them back inside a case like that the moisture has nowhere to go. I use the Lowepro cases as the fabric is breathable, and don't do things like stuffing damp lenses back inside and zipping it up - if I've been out in the rain then the camera and lenses spend a few hours in a warm, dry room before they're packed away.
I would still get a quote from a pro for dealing with it as it will affect the images eventually and the bill is likely to be less if you have it dealt with sooner. Plus they should be able to straighten the filter ring for you (they'll probably have to do this to get inside).