I have acesss to a public server in which i have read/write permissions. Only problem is that I don't have access to sql on the server. So my question is, is there any way I can either install sql remotely, or access it when I only have limited read and write permissions within the server.
No, that's definitely impossible.
No you can't install a program with limited read/write permissions. You need permissions to install packages on the machine.
You could try to upload a binary to you home directory, but I don't know if this would work.
Yes. You can do that if you can connect the system throught remote desktop and also you need administrator rights to install it.
You need administrator rights, or specific permissions to install software.
There are cloud services which allow you to install any software you want for your server, and allow you to open it up publicly. One such service is heroku.