1. In order to build something on the planet - should I be able to store the amount of resources?

For instance - I should build something that costs 400.000 metal, but I have storage capacity only for 100.000 on the planet X.

I can deliver the necessary amount from the planet Y - should I still build the storage that is sufficient to store all amount?

2. If I have my fleet bringing back certain amount of resources, but my storage is already full, what's gonna happens with the resources from the ships?

Storage only affects production. You can have as many resources as you want on your planet without adequate storage and the only effect is your mines stop producing.

This also why you should never build storage on moons.

Exactly. This is also why, when you have buldings or researches that cost 15-20 Mill of something, it would be smart to transport it from another planet instead of keeping on upgrading those storages. But, you can do that too if you want.

Thank you guys for answers, tis really helping!