I don't know the math needed to set up or solve this problem. If this isn't the right resource, I welcome any suggestions on where to go.
We have 55 employees. 26 are needed to work shifts on a given day.
There are 4 training sessions throughout the year requiring employee attendance.
Of the 55 employees, 22 need to attend only 2 of the 4 sessions in a year. The remaining 33 employees need to attend 3 of the 4 sessions in a year.
In addition to the 26 shifts noted above, there are 4 overnight shifts from the day before ending the morning of training as well as 4 shifts from the day before that end well after midnight on the day of training. Ideally, employees working these shifts would not attend training after working.
Do I have enough employees so that everyone can attend the required number of sessions?
Secondary question: Do I have enough employees so that everyone can attend the required number without having worked an overnight shift or a shift that ends after midnight on the day of the training session?
Tertiary question: Do I have enough employees so that no one who attends training has to work a shift either the night before (overnight), a late evening shift the night before (ending after midnight) or a shift on the day of training itself?
I presume you mean that you need 26 employees at work, not in training, while the training is going on.
In that case, your answer is no. Think of it like man-hours, but instead you have man-sessions. By your requirements, you need $22 \times 2 + 33 \times 3 = 143$ man-sessions of training. But during each training session, only $29$ employees can attend, so at most you can get only $4\times 29 = 116$ man-sessions. That is not enough.
You might be able solve it with more employees, but I think the easier solution is to add at least two more training sessions. (Yes, numerically one more session is enough, but the reality is you will never be able to make it work.)