For the first question,
1. 我经常参与母校举办的中国传统节日的活动。【答案:语法对,意思对,自然】
it's more common to say 中国传统节日活动 rather than 中国传统节日的活动
2. 我经常参与母校举办的活动过中国传统节日。【答案:语法对,意思不对,不太自然】
It should be 我经常参与母校举办的活动, (comma) 过中国传统节日。
This sentence is still natural, but it has ambiguous meanings, the festival activities might or might not be held by the school.
3. 我经常参与母校举办的活动,也会过中国的传统节日。【答案:语法对,意思不太对,自然】
This means the acitivities are not (necessarily) held by the school.
For the second question:
some examples,
我听新闻的时候,把作业做完了。(two actions at the same time)
我每天都打扫卫生,听歌,玩游戏...... (different actions might or might not happen at the same time)
I would not suggest you to analyze verbs the way you are doing now. It's not easy to list all the possibilites of sentences with multiple verbs.
Verbs can be in subordinate clauses, or transformed into adjectives(他吃过的盐比我吃过的饭还多), etc.