> Ideally, for smaller screens it would split across multiple lines.

C'mon, nobody deserves to be subject to automatic TeX expression breaking. Even careful hand-tuning with help from the breqn package can often look pretty bad.

> Two commands I used all throughout my thesis are:

At the risk of turning this into TeX StackExchange, I just want to point out that these kerning manipulations are entirely possible in MathJax. The problem is that, maddeningly, they chose to have non-math commands like kern and lower work exclusively in math mode, and not outside one like in a hbox. Therefore, if you change your second command to something like this:

  $a \kern-1em\lower0.4em\hbox{$\scriptstyle \rightarrow$}$

then it will work in MathJax. Of course, this would then break in regular TeX because you're not supposed to use kern and lower in math mode. I keep myself sober by remaining optimistic about a potential overhaul in the MathJax latex processing code.

In the meantime, I think we would benefit from a stopgap service that renders these things in an actual TeX engine running on a server into high-res PNG (easy) or SVG (doubtful). There's been quite a resurgence in effort to make latex rendering into a scaleable web service lately. I'd love to collaborate with someone from, say, ShareLaTeX on that, if possible.