Apache mod_rewrite fake pushState directory mapping - apache

Question

If have played around a lot with mod_rewrite rules in my httpd.conf file. Regardless of my research i haven't been able to get a couple of things working.
This is my file structure:
/
-index.php
-app.php
/css
-style.css
/js
-script.js
The server should either serve the index.php (home page) or app.php (single application page). Both the script and style files are included in both php files.
Goals
My domain domain.com should serve the index.php, the address bar should show www.domain.com
(This seems to work already, per default.)
The subdomain domain.com/a should be changed to domain.com/a/ if necessary. This domain should serve the app.php file.
(This is already working to an extend. One problem is that the relative links inside app.php are wrong, because the file "thinks" it is in a subdirectory instead of root. This I would like to change)
Anything after domain.com/a/ e.g. domain.com/a/user/10 should stay in the address bar and serve the app.php as usual. Ideally, to preserve relative links again, the file should "know" it is in the root folder.
(This is in order to support a "fake" pushState server) EDIT Clarification: Everything after the /a/ will be interpreted by my Javascript app. When the client clicks a link like domain.com/a/user/10 there will be no extra request to the server.
Bonus
Add trailing slashes to all URLs except the root url.
e.g. turn domain.com/a/user/10 into domain.com/a/user/10/
Add www to URL in case it is missing.
What I've tried
-add www
RewriteCond %{HTTP_HOST} !^$
RewriteCond %{HTTP_HOST} !^www\. [NC]
RewriteRule ^ http://www.%{HTTP_HOST}%{REQUEST_URI} [R=301]
-add slashes
RewriteCond %{REQUEST_URI} !(/$|\.)
RewriteRule (.*) %{REQUEST_URI}/ [R=301]
-redirect /a/
RewriteCond %{REQUEST_URI} ^(/a/)$
RewriteRule ^ /app.php
Any pointers or help are greatly appreciated! Thanks.
Edit
I have used this tool http://htaccess.madewithlove.be/ to test some conditions.