On that basis, I think the most likely permutations are as follows:
1. If Utd win the FA Cup, Liverpool win the EL, and Liverpool finish 7th:
Utd and whoever finishes 6th (or 5th, if Utd are overtaken) between us and West Ham go into the EL group stage.
2. If Utd win the FA Cup, Liverpool win the EL, and Liverpool finish 8th:
Utd and whoever finishes 6th (or 5th, if Utd are overtaken) between us and West Ham go into the EL group stage. The other side (finishing 7th) goes into the EL qualifying.
3. If Utd win the FA Cup, Liverpool lose the EL, and Liverpool finish 7th:
Utd and whoever finishes 6th (or 5th, if Utd are overtaken) between us and West Ham go into the EL group stage. Liverpool (finishing 7th) go into the EL qualifying.
4. If Utd win the FA Cup, Liverpool lose the EL, and Liverpool finish 8th:
Utd and whoever finishes 6th (or 5th, if Utd are overtaken) between us and West Ham go into the EL group stage. The other side (finishing 7th) goes into the EL qualifying.
5. If Utd lose the FA Cup, Liverpool win the EL, and Liverpool finish 7th:
Palace and whoever finishes 5th go into the EL group stage. Whoever finishes 6th goes into the EL qualifying.
6. If Utd lose the FA Cup, Liverpool win the EL, and Liverpool finish 8th:
Palace and whoever finishes 5th go into the EL group stage. Whoever finishes 6th goes into the EL qualifying.
7. If Utd lose the FA Cup, Liverpool lose the EL, and Liverpool finish 7th (or 8th):
Palace and whoever finishes 5th go into the EL group stage. Whoever finishes 6th goes into the EL qualifying.
I think this illustrates (if I am correct) that whilst Liverpool's final league position is of relevance to us, whether or not they win the EL is of no relevance - eg scenarios 2 and 4 give the same answer. (But whether or not Utd win the FA Cup is absolutely of relevance.)