A Trainee at most belongs 1 trainee project at a time

That can be shown through a constraint saying for a given trainee the dates of the projects never overlaps.

In OCL the constraint can be written :

context Trainee inv:
 self.ProjectHistory->forAll(h1,h2 |
                             h1<>h2 implies (h1.dateCompleted < h2.dateStarted) or
                                            (h1.dateStarted > h2.dateCompleted))

also having :

context ProjectHistory inv:
   self.dateStarted <= self.dateCompleted

supposing dateCompleted is set with the current date while the project is on going, and we know the project is on going through an other way.

If dateCompleted values 0 while a project is on going :

context ProjectHistory inv:
  (self.dateStarted > 0) and 
  ((self.dateCompleted = 0) or (self.dateStarted <= self.dateCompleted))

context Trainee inv:
  self.ProjectHistory->select(dateCompleted = 0)->size() <= 1

context Trainee inv:
  self.ProjectHistory
    ->forAll(h1,h2 | 
             h1<>h2 implies 
               if h1.dateCompleted = 0 then
                 h1.dateStarted > h2.dateCompleted
               else 
                 if h2.dateCompleted = 0 then
                   h2.dateStarted > h1.dateCompleted
                 else
                   (h1.dateCompleted < h2.dateStarted) or 
                   (h1.dateStarted > h2.dateCompleted)
                 endif
               endif
             )

Out of that to use an association-class is a right way.

Over time, Trainees may take more than one project.

so any number, the multiplicity 0..1 must be 0..* or the shortcut *.

The multiply must be 0..1 if that rule and the previous are replaced by (over time) a Trainee at most belongs 1 trainee project

A section may run more than one project.

so any number, the multiplicity 1..* must be 0..* or the shortcut *

A Project can have may trainees

may is visibly many so the high number is not 1.

Just having a project have many trainees probably implies the minimum number is ,1 but with can have the minimum number is 0 rather than 1.

So finally the multiplicity is 0..* or the shortcut *, rather than 1..1

How would previous projects get retained?

they are thanks to the multiplicity 0..* rather than 0..1, the trainee does not have a relation to only the ongoing project if exist

To summarize :

Example of object diagram :

you can check all the rules are followed.

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In several case the name of your attributes contains the name of the class, for instance projectName and traineeId, this is useless and I recommend you to remove the name of the class in the name of the attributes.