Wow, that's a really tough question to explain considering I'm not a pro in teaching Chinese, althrough I'm a Chinese, but I'll give it shot:
1. Your got it right on the mark.
2. Sure you can replace "以" with "根据" meaning "according to", but "为了" is clearly not a good option in this case, for "为了" means "for". Why "以" is the best bet? Well, with their corresponding translation, you can tell:
1) (以)With her results in the exam, she can definitely go that famous university.
2) (根据) According to her results in the exam, she can definitely go that famous university.
You can also see that there is a tiny change of meaning with "根据", so "以" is definitely the best.
3. ""可以以"" is as a matter of fact two words "可以” and “以” here. Just as the translation shows, "可以" means "can", while "以" followed behind means "with".
4. No, "以" alone can never be a verb, so it's not "compare", it is more like the "to" in "compare to". If only you can say the sentence in this way "He always compares himself "with" the red wintersweet to show his revolutionary standpoint and determination.".
5. You got it right again!
6. Are you sure the translation is correct? Because from what I see here, the translation "She criticized her husband for making all kinds of excuses" has a different meaning from the original text, which is "她因为丈夫找诸多借口而批评他。” I would rather translate the original text "她以种种的理由贬责自己的丈夫" into "She criticized her husband with all kinds/sorts of excuses". So you can easily tell "以" is "with" here.
7. "以" also means "with" here, if I change the structure of translation a little bit, it may help you to understand: "The debaters from the opposite team attacked his opponent with an argument, forcefully, and convincingly". So the use of "以" here is actually the same as No.2, 3, and 6.
8. Well, the translation is actually not a literal one, it just got pretty much the gist. In fact the corresponding Chinese of the translation here is "革命先烈‘的’(equals "of")彪炳千秋的光辉业绩激励我们不断向前。If you translate the original one literally, it will be "The evolutionary heroes inspire us to move forward forever with their great achievements."
9. "以" means "according to" just like No. 2, and you can replace it with "with" as well.
10. Like No.8, it's not a literal translation, which should be "When debating, one must convince others with reasonable facts."
OK, in a word, in most cases, when you are refering to "with” you can turn to "以", only a small number of cases, when you talking about "according to", "以" is your option.
All right, quite a lot to type, here. More importantly, you got it?