Login Problems with different Site URL and Home URL ( Social Login)

Dear www.ads-software.com community,

we are running a WP 4.2.2 based Website, with the main plugins Buddypress 2.3.4, Rating-Widget 2.5.8, Geo Mashup 1.8.4 and OA Social Login 4.6. Our theme is the Mustang theme von WebMan! (kriesi.at). Several custom codes snippets have been implemented and tested on a development server.

All functionalities have been working on the development server, especially the admin-login and Social-Login functions.

Now to the problem, since we transferred the page to our live server, some things don’t work as they are supposed to.
The main difference is, that on the live server we use a subdirectory for the wordpress page (it is not directly in www), where we point our domain (www.startouched.net) to.

The site url is: https://peaceful-seedling.com/startouched/
The home url is: https://startouched.net

For the Transfer, we followed strictly the tips of: https://codex.www.ads-software.com/Changing_The_Site_URL

All database-entries and strings that point to the old development server direction have been altered to the new site url!

the htaccess file located in peaceful-seedling.com/startouched/ looks like this:

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
Header set Access-Control-Allow-Origin "https://startouched.net"
</IfModule>

The wp-config doesn’t have any cookie domain defined or restrictions there.

Still the following problems remain:
1.) Login into admin from https://startouched.net/wp-admin does not work, no error message (even with debugging) appears. I found out, that no cookie from peaceful-seedling.com will be created
2.) Login into admin from https://peaceful-seedling.com/startouched/wp-admin works, a cookie from peaceful-seedling.com is being created. Changing admin settings works without a problem.
3.) BUT: as soon as we visit the main blog https://startouched.net/blog/, the admin-user is not logged in anymore and can’t e.g. create comments. The cookie has been deleted
4.) Logging in with Social Login (Facebook and Google+) works, User accounts are created and users can create new posts and comments. This time a cookie from startouched.net is being created
5.) BUT: Logout of user by calling the peaceful-seedling.com/startouched/wp-login.php?action=logout does not work, the cookie won’t be deleted and the user remains logged in

I hope I could describe what bothers us: Since the transfer to the live server with a different home and site URL, we have those problems. But we can’t figure out, what causes them.

Hopefully anybody of you has some hints what we could try to make the site work properly again.

The only solution I have right now, is getting some new webspace where I can use the root directory (www) and don’t have to use a subdirectory.

Thank you very much!
Markus