About pronunciations:
Shichi/Nana and Kyuu/Ku and Shi/Yon
Both are all correct, this is a matter of taste, but in some expressions you have to use one in particular.
Japanese will use either shichi or nana.
People usually use kyuu except in some words like kugatsu as said by Sqrtbottle.
Also Japanese would rather not use shi as it has the same pronunciation as the shi in Shinu which means death, but both are perfectly correct and are a matter of taste. I know several Japanese who use shi.
About the meaning of numbers:
It is wrong to say the Chinese root ones are more commonly used.
This is just a different usage.
Basically if you start learning Japanese you should start by learning:
いち に さん し/よん ご...
Those are mostly used for mathematics and calculations.
But for speech, there are a lot of different counting systems. (to count things)
The most basic one is like this:
ひとつ ふたつ みっつ よっつ ...
You are supposed to use it for counting things, but this is not as simple as that. If you start learning, only knowing both is ok. But to understand daily Japanese you need to learn a lot of different counters:
いっぽん にほん さんぼん (counter for long stuff like bottles, straws etc ...)
いちまい にまい さんまい (counter for thin stuff like sheets of paper)
いっぱい にはい さんばい (counter for cups of drink)
There are a lot of those.
So basically, they all have their own usage. If you don't know the special counters it's ok, just use the basic version (ひとつ ふたつ...), your grammar will be correct. But using special counters is always better, it lets you build shorter sentences and will sound more Japanese.
Don't forget, you are not supposed to count stuff with いち に さん.
Let's do some examples:
いちたすいち : 1 plus 1
いっぱいちょうだい : give me one cup
As you can see in the second one, all you can see is the counter for cups and choudai, so implicitly you understand that we are talking about cups. One last example:
3まいたりない : literally means "I lack 3 thin objects", but depending on the context, you could be printing things in the office, people will understand it like: "I need 3 more sheets of paper to finish my print", this is how this sentence would be translated and how Japanese will understand it. Saying [紙]{かみ}がみっつ[足]{た}りない might be grammatically correct but it doesn't sound good.