Some minor wordpress modifications.

General information for the website: Wordpress WooCommerce site
Kind of development: Customization of existing website
Description of requirements/features: 1. Complete integration of social site user registration [IE using Facebook,Google+,Twitter to register on my site. Acquiring email address & names.

2. Automatically sending registered user information to mailchimp.

3. Proper setup of Wc3 Total cache with Cloudflare(free) and MaxCDN(paid) + implementation of suggestions from Yslow & Google speed-test to maximize website loading speeds & mobile useability.

4. Fix up the layout of http://www.lazulu.com/my-account/ so it looks nicer.

5. Integration of my logo throughout website and removal of any mention of WordPress throughout website -backend &frontend.

6. Fix: Featured images aren't displaying correctly between shop area & blog area. And not at all on mobile.
http://www.lazulu.com/shop looks fine here.
http://www.lazulu.com/blog/ abit too small. Not displaying on mobile at all?
http://www.lazulu.com/lifestyle/ abit too small. Not displaying on mobile at all?

7. Fix: Facebook comments seems to be broken on mobile?

8. Fix up photo contest plugin so its more attractive/user friendly on mobile
http://www.lazulu.com/selfie/

9. A highly visible animated icon to replace the red + icon in top right corner that opens an extra "menu" See attachment.

10. User profile pages: For some reason author names don't link to profile page or display "avatar", Or option to even upload avatar, Likely because of Woocommerce plugin I'd imagine. See: http://www.lazulu.com/photos-video-from-coachella-2015

Anyway I'd like for those options to be available again, if not very difficult.
Note: Avatars display correctly in forums: http://www.lazulu.com/forum/community

11. Preferably modifications aren't broken when updating theme/wordpress core/plugins =)

. Any other suggestions?
Specific technologies required: css,html5,wordpress,php,ajax
Extra notes: