Will it be legal to use a small part of an existing big trademark-name for a domain-name?

It is always better to use original and coined names when you are registering a new trademark. It gives the assurance of a seamless registration process and it would prevent any unwanted infringement cases against you. Sometimes using famous trademarks as a prefix or suffix can end up in expensive lawsuits. And as it is for a domain the chances of someone filing an infringement case are higher.