Seems like POST values are lost when .htaccess RewriteRule used. GET values are OK. How to fix?

The [R] flag will incur a redirect. And user-agents issue a redirect as GET request. There is nothing that can be done if you really want to shorten URLs down to the / root path.

You could however block POST requests specifically from being rewritten/redirected:

RewriteCond %{REQUEST_METHOD} !POST
RewriteRule ^index.php / [L,R=301]

You could try using [L,R=307] instead. 307's must not change the request-method according to the spec, but I don't know how browser implemented 307.

But the root of the problem is the use of <form action="____/index.php" ...

Just leave the action empty to POST to the current url e.g.

I'm using something like:

<IfModule mod_rewrite.c>

RewriteEngine on

RewriteCond %{REQUEST_URI} !^/(css|images|js)/

# don't rewrite existing files, directories and links

RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-l


# rewrite everything else to index.php

RewriteRule .* index.php [L]

</IfModule>

And its working for all requests, rewriting it via index.php file. If you need to redirect 301 (which stands for Moved Permanently code) check out this question: Is it possible to redirect post data?