The nouns 早さ and 速さ are derived from the adjectival verbs 早い and 速い, which both can mean both "fast" and "early". Being derived from adjectival verbs, they are native Japanese words.
On the other hand, 速度 is a Chinese-derived word, which means "speed" or "rate" (and doesn't mean "earliness").
The kanji 早 is usually used for the sense of "early" and 速 usually for speed, although these concepts overlap a little. (E.g. 早いですね doesn't need to be "you are early", it could be "wow, you were fast" in the sense of "you got here earlier than I expected".)
For the question about typing speed, you can interpret タイピング to have the status of a Chinese-derived word in the following sense:
Chinese-derived nouns (which are usually two-characters long) can form compound nouns simply by concatenation (without a joining particle like の). E.g. 自宅教育 "home-schooling" from 自宅 "home" and 教育 "education".
On the other hand, Japanese-derived nouns are usually connected to other nouns with の.
Thus we get
which concurs with your findings on Google. (Although you shouldn't trust Google counts!)
Here ○ and × indicate what sounds most natural in the sense of the above explanation. Of course, you can always combine two nouns with の, so タイピングの速度 is grammatically correct as well.