I bought the iPhone application, do I need to buy again the application for my iPad?
You can use the iPhone applications on the iPad, you won't have to pay again for it but the app is not optimized for the iPad and is not taking advantage of the big screen (it's not a 'universal' App). We also have a separate application optimized for the iPad called 'Daily Connect for iPad'. Because it's a separate app you'll need to purchase it independently from the iPhone app. All entries will be synchronized on both apps.
My partner already paid the iPhone/Android App, do I also need to pay to download it on my iPhone/Android phone?
If you are both sharing the same iTunes/Google Store account, you only have to pay once. If you're using a different account, then you'll have to buy the App for each phone.