Make sure you’ve redeemed the key which came with the standard edition and if you have multiple accounts, make sure you login with the correct one. Sometimes it may take up to 72h before the system properly updates and your license goes from starter->standard.
You do not have to purchase Reaper of Souls or Rise of the Necromancer. These are optional. But, you’ll be missing the best part of the game without RoS. The Necromancer is optional.
All you need to do is register your License Key by logging in to your Blizzard Account online, as Kirottu suggested.
RoS is not required to create a seasonal character. Of course without it one can’t really benefit from seasons but nothing stops from creating a seasonal character without it.
Even if it does, Adventure Mode is unlocked by completing Act V and D3 does not have an Act V.
However, Adventure Mode is automatically unlocked for Seasonal Characters, so… I don’t know either. (I pre-ordered RoS and have no way to test this).
Adventure Mode, Rifts, Greater Rifts, Bounties and Challenge Rifts are ROS only which means that the vast majority of the Season Journey tasks are impossible to achieve without it.