Can I reinstall iPhone applications already purchased / bought on iTunes for free?

Yes. Once you purchase an iPhone app it can be reinstalled for free. You can check to make sure you already own an application by getting a list of apps you've purchased. When reinstalling an app you already own you will be prompted with a window asking if you'd like to re-download the application for free. Apps already stored in your iTunes library can be reinstalled to a USB-connected device by selecting the app to be synced in iTunes.