hodně on it's own just means a big/large amount, like много if I guess correctly you will understand Russian from your nickname. If you want to compare something, you use mnohem or o hodně which means something like by a big/large amount similar to гораздо.
The o is something like the by above.
Thank you. Yes, I'm Russian. So it is like some figure of speech in Czech: "o hodně" = "mnohem"? Because I still can not build straight parallels to English or Russian with your examples. Only to accept that it so. In Russian it is possible to use "много лучше" in place of "многим лучше". And the form "о много лучшем" has diffent meaning and somewhat unusual, it is used in some specific conditions. In English as far as I know "by far" is used for the "superlative" comparing: "is by far the prettiest". If you had some other examples, which could give the solid analogue of this sentence to English or Russian, it would be great.
I really wouldn't formulate it like that. It is not a different form of the word. Yes, you are right for mnohem.
But o hodně is just a regular use of the o preposition that has the comparative meaning here. You can use it with any other quantity.
Vyhráli jsme o pět gólů.
Toto auto stojí o pět tisíc korun víc.
Můj bratr je o 5 cm vyšší než já.
Tato řeka je o metr hlubší.
Tato košile je o 20 % levnější.
Budu muset chodit do školy o jeden rok déle.
There is a way to get a bit closer to complete understanding of that "o" preposition usage. Just keep in mind that Czech language has many and many more old-slavic archaic words and constructions than modern Russian does. Do you remember "зверь о семи рогах" or something like this? So we have straight analogues here even though we forgot them.