Yes, she was able to figure it out.
Lord Hoster Tully, while slipping in and out of consciousness, spoke Tansy along with other words and phrases like “blood”, “forgive me”, “be a good wife”, and “you will have others, sweet babes and trueborn”.
Catelyn, at first, thought of some commonfolk woman named Tansy with whom her father had relations but then realized that he was looking for forgiveness from his younger daughter. She also penned a latter to Lysa telling her how sorry their father was.
As for second part, while it is not confirmed in books, I believed Cat knew that as well. She figured it out that Jon Arryn needed a young bride who was fertile to bear him sons and her father placed the condition of marriage with Lysa in exchange of his support.
If Lysa was with child before her marriage the only person who can be the father could be Petyr.
If she had lost a child before, that might explain Father’s words, and much else besides. Lysa’s match with Lord Arryn had been hastily arranged, and Jon was an old man even then, older than their father. An old man without an heir. His first two wives had left him childless, his brother’s son had been murdered with Brandon Stark in King’s Landing, his gallant cousin had died in the Battle of the Bells. He needed a young wife if House Arryn was to continue, a young wife known to be fertile.
“You made him take her,” she whispered. “Lysa was the price Jon Arryn had to pay for the swords and spears of House Tully.”
Lord Jon might wed Lysa to bind the Tullys to the cause of the rebellion, and in hopes of a son, but it would have been hard for him to love a woman who came to his bed soiled and unwilling. He would have been kind, no doubt; dutiful, yes; but Lysa needed warmth.
Image source – Winter is Coming