How can both 过 (past actions) and 会 (future actions) turn up in 下过雨以后，空气会好很多?
I learn that "过" is used for past actions, "会" = "will" (for future actions). So, can both 过 and 会 turn up in this sentence? 


Your understandings of both characters are correct. But here in this sentence, the usage is a bit similar to "subjunctive mood" in English. 

The sentence basically means: "After raining, the air would be much better." You can understand it as "If it rained, the air would be much better."

But honestly, the concepts of "tense" and "mood" are from Roman languages. In Chinese, there is not a prohibition on putting two tenses in one sentence. 

For example (I just made up the sentence, might not be perfect): 

“我刚试过了苹果的新产品，一发工资我就买！” ("I have just tried Apple's new product, I would buy it on my next payday." )

In English, you have to put the words into the right tense, but in Chinese, it's almost the same for "present perfect" and "simple past".



Here are the standard usages of the two characters in Modern Chinese Dictionary 《现代汉语词典》.

过【·guo】 

1 用在动词后，表示完毕：吃～饭再走｜杏花和碧桃都已经开～了。

2 用在动词后，表示某种行为或变化曾经发生，但并未持续到现在：他去年来～北京｜我们吃～亏，上～当，有了经验了。

Comment: Two meanings are similar. Here is the first one. "过" has many other meanings under other tones. This one is "neutral tone", but in oral usage you can use “falling tone” [guò] to emphasis the status of "finished".



会2【huì】

5 （助）助动词。表示有可能实现：他不～不来｜树上的果子熟了，自然～掉下来。

Comment: 会 has more usages than I ever noticed. This one means "will" but without "a will". When someone says “我会去做的。”, it is the same usage but implies "I will do it without you saying." "It's in my plan already."



I'm a native Chinese but inexperienced in explaining my mother tongue. Feel free to ask follow-on questions. I hope my answer could help you.