how to make a db connection created through ADO available to all methods within and outside the unit

I'm not a python dev by any means but something like this I think is what you are looking for:

connectSting = ""

AConnection = ADO.CreateADOConnection()

def GetConnection():
  if AConnection.Connected == False:
    Log.Message("Connecting")
    AConnection.ConnectionString = connectSting
    # Suppress the login dialog box
    AConnection.LoginPrompt = False    
    AConnection.Open()
  elif AConnection.Connected == True:
    Log.Message("Connected")
  return AConnection
  
def test():
  connection = GetConnection()
  connection = GetConnection()

Then if you want to use that in other script units see: https://support.smartbear.com/testcomplete/docs/scripting/calling-routines/declared-in-another-unit/python.html