Don't be surprised if console mods do cost money. I'm not saying they will, but they very likely could. No there will not be multiplayer mod support on the ps4 or xbone. Even if there are multiplayer mods available they will require server access. Since these mods will use user created servers it will be highly unlikely that microsoft or sony would allow you to connect to "unauthorized" servers, not to mention the fact that this would also introduce crossplay between all three platforms; microsoft and sony will not like this. Mods on the ps4 and xbone will be very, very limited. Anything that replaces textures with higher resolution textures, better lighting, enb, better character models, custom animations, and anything that requires a script extender can almost certainly be ruled out. So you'll basically be limited to standalone follower mods, weapons, armor, and maybe some quest mods. But a lot of quest mods will require script extenders, so being generous maybe 50% of quest mods will be available. So to put that into perspective; my Skyrim has about 150 active mods. Only 13 of those would work on a ps4 or xbone if the same ideas are applied to a hypothetical skyrim on current gen.

Supposedly any mod available on console will need to be approved by Bethesda beforehand. Demanding graphics mods and the likes likely won't be on console, so no worries.