A First Lady would not plan a state funeral. It would be planned by the decedent's family, alongside the division of the Department of Defense with purview over state funerals, the Joint Force Headquarters National Capital Region.
Leonard Carnehan's defense team wasn't able to test the bullet in Cooper's skull, yet there is no mention of any testing on his gun to see if it had been fired or even if a round was missing, or whether he had powder residue on himself. These details would help to make Olivia's trust in him more plausible.
Fitz went through law school, and became a Rhodes scholar, and his intelligence is often trumpeted. Yet he is convinced of Jake's guilt in his son's murder by some pretty slim evidence: a photo of Tom and Jake on a bench, photographic evidence of Tom stealing the vial, and Tom's confession. A lawyer would know this evidence doesn't touch Jake, especially given that Rowan, Tom's former boss at B613 and a competitor with Jake for that position, got the confession. Unless Fitz somehow thought Jake had only just recruited Tom into B613 he would have to realize Tom was Rowan's agent, an agent Rowan placed close enough to him that he could kill Jerry. Even if Fitz thought Jake had taken total control of Tom, Fitz would still be stung by Rowan's initial act of planting a mole in the Secret Service, and likely trust him less, especially given his usual machismo. Fitz's initial trust in Rowan was founded on the mistaken belief Maya Pope was solely responsible for the murder, and so when evidence against the agency Rowan created turned up, Fitz would not likely be so blinded by jealousy of Jake that he wouldn't start asking questions about Rowan.