I'm not one of the JSXGraph developers, but I'll help as much as I can.
(1) I'm not sure what you mean by "the two intersections collapse in one". In this fiddle, when the tangent is vertical, it behaves properly and doesn't disappear.
https://jsfiddle.net/mbourne/mwp64h3r/
(2) I'm not sure how JSXGraph applies options.js, so I will leave that to Alfred to respond to. In general, my understanding is JSXGraph makes use of the defaults in options.js, and the developer can change those by setting options when creating elements.
(3) To make the value invisible, you just add withLabel: false when creating the slider.
var n = brd.create('slider',[[0.1,1.4],[1.1,1.4],[3,3,20]],{name:'n', snapWidth:1, withLabel:false});
See it in action here:
(4) Presumably you are using MathJax to display your LaTeX? Are you following the steps in this page?
https://jsxgraph.uni-bayreuth.de/wiki/index.php/Using_MathJax (The MathJax part was really slow for me when I looked at this page just now and the fonts didn't load properly. Use the CDN version of MathJax.)
If you can create a fiddle with what you have tried it would be better - we can take a look at why it's not working.
(5a) The answer to the increments for a slide is given in my answer (3). "snapWidth" is what you need.
(5b) I recall there is some syntax for changing slider colours, but I couldn't find it.
Here is one way to do it. I got the required IDs via the DOM inspector. (This is dangerous as the IDs could change when you add other elements - it's just to show you one way it could be done.)
// This changes the initial circle and segment colours:
document.getElementById("jxgbox_jxgBoard1L19").setAttribute("stroke", "#f0f");
document.getElementById("jxgbox_jxgBoard1L21").setAttribute("stroke", "#f0f");
// This changes the colors as you drag the slider (otherwise, they just go back to being black)
n.on("drag", function() {
document.getElementById("jxgbox_jxgBoard1L19").setAttribute("stroke", "#f0f");
document.getElementById("jxgbox_jxgBoard1L21").setAttribute("stroke", "#f0f")
});
You can see it in action here:
The highlight colours still default to black. I'll leave that as an exercise for the reader :-)
(6) This appears to be related to (4).
(7) My advise is to put your radio button (for on-off) outside of the JSXGraph construction (that is, next to the graph but using normal HTML). It's usually neatest, especially if the user wants to zoom in-out on the graph (whereby any buttons or other interactive elements can disappear).
I hope it helps