How many times can American Tourist return to the United Kingdom during the same 6 month period?

If, in the middle of my trip, I fly a few days in Italy and want to come back a few more days in the UK before returning home, can I go back to the UK?

Yes, however, as always, the immigration officer may refuse entry if you do not convince him that you are an "authentic visitor".

And if I can go back to the same six-month tourist limit – how many times can I go back to the UK again (eg between April and October)?

There are no limits. You can return to the UK as many times as you can, while persuading the agent every time you are a genuine visitor. This will become more difficult as the time spent in the UK increases. This will also tend to be more difficult if the indicated duration of your intended stay is longer or if your projects are more indeterminate.