Learn Russian: Он легко решил все задачи.

In Russian, a задача is a task, an objective, or a specific problem to be solved, like a math puzzle or a work assignment. A проблема refers to a difficulty, trouble, or a negative issue in life. Since he successfully completed them, they are tasks or exercises, so задачи is the natural choice here.

Could we use the imperfective verb решал instead of решил here?

You could, but it changes the meaning. The perfective решил means he successfully completed solving all of them, putting the focus on the final result. If you used решал, it would mean he was in the process of solving them or spent time working on them, but it doesn't guarantee he finished them.

What case is все задачи in, and why?

It is in the accusative case, plural, because it is the direct object of the verb решил (the things being solved). For inanimate feminine nouns like задача, the plural accusative form looks identical to the nominative plural (задачи), and the modifier все (all) also matches this inanimate plural form.

Is все here pronounced with a "ye" or a "yo" sound?