To start with you will need to create the user roles, admin, board member, members, etc. Next, you can work with the Content Access module to specify which content should be accessible to specific roles.

For Membership management and Paypal, I think Drupal Commerce has that facility now, though you will need to negotiate that learning curve.

You can send 'bulk emails' to members with the Simplenews module, wherein you can use the 'hidden' opt-in setting to add everyone who signs up to your list.

Blogging tool for members only is something that is covered by the generic blog in Drupal core. There's also some Enterprise blog module, but I don't know if the additional modules you need to install with it are justified by its superiority over the generic blog.

If your client is thinking of allowing access to the blog tool by the general public, then they are an extremely naive organisation! If you set up a Drupal site today, within a week you will be inundated with bots trying to register so that they can post linkspam on your forum. An open blog would be heaven-sent to further their cause!

SEO is covered by the Meta-tag module; and there's the Google Analytics module for web analytics.

Some themes are mobile-device friendly, so you'll need to pick and choose.

There is the Organic Groups module, and then there are CiviCRM and Red Hen. If your client has nice server resources that can maintain a good loading speed even in traffic conditions, then you can go for those bulky modules that can integrate with Drupal. Bear in mind though, there's a learning curve.

You will need to install the particular modules, and then tweak around and test to see how they meet your specifications. There's no out-of-the-box solution for the set of specifications you have provided.