Can you do IRT, CFA and/or Rasch if your data has items with large ceiling effects (i.e., no one was choosing the last two options of a 5-point likert scale)? Is it appropriate to do?
If you have large ceiling effects, there's nothing you can do about it. With the more sophisticated measurement models, you'll be able to see that the measurement quality deteriorates for those with high scores as you have little information to differentiate them.
If the test is to detect whether an individual knows certain things, ceiling effects are fine; they show those individuals know those things. However if your goal is to measure across the sample under study, then there's no solution.
The nice thing about Rasch/IRT is the process of running and diagnosing the model will reveal these problems to you. EFA can give you similar information, but information about ceiling effects is usually hidden behind many layers; EFA emphasizes other features of the data.
So all you can do is measure with the caveat that for those of high ability, you're struggling to differentiate them properly.