Is it legal to build a separate blockchain with our own coins/tokens and miner-code to mine those coins/tokens (not a private one) based on ethereum open-source code ?
Yes, you're welcome to copy the code and run your own Ethereum-like network. The most famous example along those lines is Ethereum Classic, although it's a bit of a special case because it starts with a version of the Ethereum ledger, and differs by not changing when Ethereum did. You can also start a new ledger, without previous account balances, and you can credit your own accounts if you like.
However, if you want your network to be secure, you will need to attract a lot of hashpower, which implies you will need people to consider your coin very valuable. If you just want to be able to use your own token, it makes more sense to run it on top of the existing Ethereum network. That way you don't have to pay for mining, and it's easy for people to buy and sell your tokens in exchange for Ether or other tokens. Unlike Bitcoin, Ethereum is designed to be used in this way, so it's much more common to put the token on the existing blockchain rather than trying to bootstrap your own.
Just to add to the previous answer, and to save on problems further down the line, "open-source" is a rather general term when applied to software licensing.
Before embarking on a project based on a fork of the codebase, it'd be prudent to understand the licenses that apply to the different parts of the code, and how they might apply to your use-case. (Note: There are several different licenses.)