For (1), I have edited info of field-theory.
For (2), moderator Alexander Gruber♦ has created a synonym and contributor YuiTo Cheng improved info of tqft.
For (3), contributor YuiTo Cheng added info and usage of conformal-field-theory.
For (5), I have edited info of mathematical-physics and I think there's still room for improvement.
I edited some questions about with mathematical-physics and quantum-field-theory these days, and I'd like to share my ideas here.
[Solved] (1) About field-theory, some users (mainly from Physics SE) wrongly use this tag for questions about physical field (gravitational field, electronmagnetic field, quantum field etc.), and some wrongly use this tag for questions about vector field. To avoid misunderstanding, maybe we could add something in tag info like this(?): Do not use this tag for questions about physical field, use [quantum-field-theory] or other tags instead.
[Solved] (2) About tqft for topological-quantum-field-theory, although tqft is an abbreviation, it's hard for users to find the existence of this tag or recall this when adding tags. (There were 6 questions with this tag before, after my edit, there're 18 now.) When you typing "topo...", tqft won't show up. Since I don't have the privilege to do so, I sincerely suggest rename this tag as [topological-field-theory] for users' convenience, and this doesn't change its meaning. What's more, the tag info should be updated.
[Solved] (3) About conformal-field-theory, there's no usage guidance and tag wiki yet :(
(4) About gauge-theory, there're some questions about gauge field theory with this tag, although they may not relevant apparently, I don't think we should add a tag [gauge-field-theory], quantum-field-theory is enough.
[Solved] (5) About mathematical-physics, this tag is wrongly used in many elementary physical questions and I deleted a lot of this tag these days, my thoughts are here. Maybe we should explain more in usage guidance.
Thanks for your kindly discussion. For our better community :)