Yes it is safe and allright to cut a way that’s a member of a relation, if there is a good reason. The editor will handle this by adapting the reation, but… (there is always a but…) the relation should be in the editor’s working space, otherwise the editor can’t handle it. The mapper has to make sure that it is.
As for the name: Cutting the way gives two sections with the same name. It’s up to the mapper to determine if the cut is worth it. I think it’s up to the data users to handle multiple sections with the same name as they see fit.
About turn restriction relations: in principle you need only one way as from member and one way as a to member. There’s no harm if this way is cut in two sections; but you might want to edit the extra section out, for neatness. I’m afraid I have not always done that when splitting roundabout ways (e.g. of two-ring and turbo roundabouts), to cater for busroutes and bicycle routes, which need different sections of the roundabout loops as members.
About routes: the editors should add the correct role in the relation to the new sections. AFAIK editors are handling this, if they have the relation in their downloaded working layer.
About mulipolygons using tagged ways as outer. This is valid for multipolygons, and it has been practiced extensively in the early days (so I’ve been told). The main problem is that highways have been used a lot for this, in mega monster tentacle MP’s, which is a major PITA even for reasonably experienced mappers who want to update the area.
In my opinion, it’s not a problem to use a section of fence as an outer for a wood area MP, provided that the fence is actually the outline of the wood area. In my experience, the fence seldom is; it’s usually placed within the wood, which warrants a separate way for the fence. But let’s say, school fenced school grounds or fenced playgrounds could be mapped as MP’s with the fence as an outer.
I have in fact done this with a few playgrounds surrounded partially by fence, partially by a hedge. And also, I have mapped roundabout centre islands outlined by an apron kerb, as MP’s tagged area:highway=centre_island, with the closed barrier=kerb way as the outer, and no inner. In fact, I have split a few of these closed-way outers, because there were sections without kerb, so I had to remove the tag barrier=kerb from these sections. No problem for the MP. JOSM handled it as it should.