Q: If I delete a song or songs from my iPhone I purchased through iTunes will it be completely deleted or just deleted from my phone?

It won't be deleted from your computer's iTunes if you have a copy of it on there - if you don't then you can copy it over via File > Devices > Transfer Purchases.

Whether you will be able to re-download from the store will depend upon what country that you are in (it's not possible in all countries), and whether it remains in the store - you are better off having your own backup copy of all of your downloads rather than relying on re-downloading