Robert uses the term "left-footer," a derogatory term for Irish-Catholics. Anti-Catholic sentiment was very common in England at this time.
Ivy uses the phrase, "A cat can look at a king," an English proverb that means even someone of low status has rights. It implies that all people have certain minimal rights by virtue of being alive. Like many proverbs, the origin is unknown.
Matthew says to Tom "I've been on a steep learning curve since arriving at Downton". "Learning curve" is commonly thought to be 1990s jargon, but the concept was first proposed by German psychologist Hermann Ebbinghaus in 1885 and the English phrase was first used, although perhaps not exactly as Matthew uses it, by Edgar James Swift in 1903.
Robert acts surprised when he finds out that baby Sybil will be a catholic. He shouldn't have been surprised though since he said himself when he and Cora found out Sybil was pregnant that they would have a Fenian grandchild.