Some airports are more efficient than others. If you have an hour connection, look at what happens if you miss the connection. For example, if you fly to Frankfurt and have to fly from there to Venice...if you are late into Venice, are there direct flights to Venice later in the day, on the same airline? Chances are you'd get put on the next flight if you miss your connection (assuming you booked it all as one ticket).
You will go through immigration ("customs" is a 10 second walk past a customs agent without stopping for most people) when you enter the Schengen Area - which is most European countries but not all of them. If you fly first to Paris, Frankfurt, or Zurich, these are all in the Schengen Area, so you go through immigration there. Your next flight on to Venice (Italy is also in the Schengen Area) is like a US domestic flight - no need to go through immigration when you arrive in Venice in that case.
However, the UK is not in the Schengen Area. So if you fly through London, you WON'T go through immigration there - you'll stay "airside" in London (go through security again) and not go through immigration until you get to Venice, when finally entering the Schengen Area.
Yes, you must get off the plane either way and change to another one on to Venice.