An electoral democracy, we choose those that make the laws. Due to the 2 party system we don't have many options for when they fail to do what they claim they will.
A true democracy would empower each of us to do as you say, but that's not what's here.
Our republic is quite corrupt due to greed and power, as well... Not as corrupt as many countries out there, mind you, but it's alot worse than it should be.
We're starting to resemble a corporatocracy in many ways these days as a result of all this.
'Republic' is about whether or not you have kings. Democracy is as opposed to autocracy, with a few other *cracys between those poles.
America is a democratic republic; the United Kingdom is a democratic monarchy. The US has a broken democratic system that does a really bad job of electing the most preferred candidate. It's almost cheeky to call America democratic.
Erm... Larger by land mass? Yes. Larger by GDP? No. Larger by number of inhabitants? No. Larger by amount of vastly different cultures that somehow get stuff like this done whole very decidedly not being a hive mind? Also no.