Yes, you can license the work under new terms to everyone or just to this one person, as you like.
You don't need to release a new version to add a license. If you did, then companies that sell licenses to use their software would have to bump the version number every time somebody bought one.
Am I free to rescind the share-alike requirement of GPL by just ...
In general, as you are the sole author, you can release your work under any license and you can change the license at your will.
Since you cannot change the licence retroactively, you cannot say "Version x.y.z which was under GPL from today is under your-needed-license".
So I think that the the simplest solution is for you to relase a x.y.z+1 ...
I am referring below to sections in the license language of LGPL_v2.1.
I believe 2 options are open to you:
a) You may distribute the entire library (all the files, even those you are not using) under Section 1 of the license.
b) When you are selectively ship only part of the library (the parts you need), then this would be considered a modification of the ...
There is no universal rule. You have to file a bug and ask.
In general, it is probably safest to assume that the most restrictive or most specific license terms apply, unless there is some indication that the author intended to dual-license (i.e. that they wanted to let you choose which license terms to follow). In this case, that means you're probably ...
Whenever you form and distribute a derivative of a GPL work, you must license that new derivative, as a whole, under the GPL.
Your code, in isolation, can be under the MIT license, but when it is combined and distributed with GPL-licensed code, that combination (probably) forms a derivative work which must be licensed as a whole under the GPL.
Note that the ...
The general consensus is the Stack Overflow answer you have linked to is incorrect; linking to a library creates a derivative work and therefore the GPL applies to the combined work. You can see some arguments in favour of this view here. Unsurprisingly, this is also the view of the FSF.
For the opposing view, see this question.