Do all Android phones allow installing new Applications?

there are some apps in the market that are only for specific phones/tablets, but for the most part they all work on all phones/tabs

 

For the most part you should be OK. I have occasionally heard of some non-US phones not having the Market preinstalled. And I know that sideloading (installing apps from outside the Market) is (was?) blocked on some T-Mobile phones. And paid apps aren't available in some non-US countries from the Market, only free ones. But if you're in the US and the phone has a Market icon in the app drawer, you should be OK as far as having access to most compatible apps.