Tom is on the run from the Irish police. When he gets to Downton he beats so loudly on the front door everyone at dinner hears him. Mary lets him in and he informs her not to tell anyone he's there. This makes no sense, if he wanted to get in unannounced he would have gone to the servants entrance, not beaten on the front door so loudly he made his presence known to everyone.
When Mrs Hughes is burning the toast, there is a different amount of smoke coming from it with every camera angle they use. Sometimes a little, sometimes a lot. It jumps all over the place.
Although there was a Drumgoole Castle once, it was no opulent edifice in 1920. The original Castle Drumgoole stood in County. Monaghan, not Dublin, and was captured by Oliver Cromwell in 1649 and subsequently remodeled as a hotel. Another fiction occurs when Mary regrets that her friend and fellow débutante, Laura Dunsany, is now homeless is equally false. The Lord of Drumgoole in 1920 had no daughters.