Since two months I use LaTeX4technics almost every day. I think this site covers some of your aspects.
You're able to write LaTeX-code on this site (supported by a lot of useful tools).
I use this service mostly because of the note system.
You can create equations or LaTeX-projects and they get stored as note if you want. If you're a user (for free) you can label your equation with a title and protect it with a password. With your user account you can access all your notes you have ever taken.
That's great because you can reuse a lot of equations with a minimum of effort and organize yourself.
This saves time when you e.g. write papers and you need recurring equations in your main LaTeX projects.
Some weeks ago they announced to build up a equation database. I think they want to build up something like a search engine for high quality LaTeX equations which gets filled by the community. The goal is to save your time by using LaTeX code snippets which other users have already written.
At the moment they don't offer hand writing recognition.
You also could use sharelatex if you want to organize bigger LaTeX-projects. But you can save only a certain number of projects for free. Furthermore for me it's not always necessary to set up a complete LaTeX-project, a thing that is a must on sharelatex.
At the moment they also don't offer hand writing recognition.