Category drop down

Following some WordPress examples, I came up with the following simple category selector. They are drop down menus that take you to one category at a time, the following code is added inside the php code widget, on a sidebar.

Question:
How to add “search” and “reset” buttons side by side, below the drop down menus, and how to make the search “additive”, like a multi selection search, as oppose to one category at a time.
For example, someone wants to search for ‘events’, on a particular day AND by ‘type’ AND maybe for a specific time, etc…

This is just an example, could be anything. I was inspired by the multi category plugins, but there are a couple of things that I don’t like about it; 1) it does not say (in numbers) what is available per category in bracket, 2) the position of the ‘search’ and ‘reset’ button is odd, 3) I try to minimise the usage of plugins, so I am nearly happy with the code below.

[Code moderated as per the Forum Rules. Please use the pastebin]