Q: iCloud

Yes - If you have Automatic Downloads enabled on the iPad. Settings>Store>Automatic Downloads>Apps - On.

You must be using the same Apple ID on both devices.

If you buy an app on the iPhone then if it's compatible with the iPad (e.g. you have the relevant iOS and it isn't one of the iPhone apps that require full phone capability) you can either download it manually on the iPad (re-downloading) or you can use automatic downloads so that all new app purchases (and optionally ibooks and music depending on which country that you are in) are automatically downloaded on the iPad.