I Am I allowed to do that or have to have pro plan, I am not mentioning about the landing page for a buzz site, but actual operational site with in rages of the free limit provided by the netlify
Yes, as long as you don’t resell the free plan, you’re fine. You can host commercial projects for free.
can I hand over someone my logins to manage my site on be half of me?
You shall be responsible for maintaining the confidentiality of usernames and passwords associated with your account. Each user must have unique login credentials that may not to be shared by multiple users. You are responsible for all activities that occur under your account.