Sentences like this are a problem. The subject is singular and the predicate is plural, and although in your example, the verb does agree with the subject ('the lesson is'), it sounds awkward. As a result, we often change them (as Elizabeth-Anne's example). I think you could also use 'exercise' or even 'PE' (physical exercise) or 'exercising': 'The last lesson is exercise/exercising/PE', which also removes the confusion between physical exercise and other types of exercises, such as 'English grammar exercises' :)

"The last lesson is exercises" The subject of your sentence is lesson. This is singular, in English if the subject is singular the linking verb must also be singular. Is and was are singular linking verbs. Are and were are plural linking verbs.

On the last class of the school day we do some light exercise

Yes you can. But is better to specify physical exercises, since exercises can refer also to study.