How to remove php extension and also redirect extensions to no extension form - apache

Question

How can I have this in my htaccess:
/page.php => redirects to /page
/page => shows the page.php
/index => redirects to root
And how to remove trailing slashes from all URLs, specially the root one?
Although, I'm not familiar enough with Apache configs, but I could find these codes:
# To remove .php extension
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^([^\.]+)$ $1.php [NC,L]
# To remove trailing slashes
RewriteRule ^(.*)/$ /$1 [L,R=301]
# To redirect /index to root (not sure if it works correctly)
RewriteCond %{THE_REQUEST} ^.*/index
RewriteRule ^(.*)index.php$ /$1 [R=301,L]
Could you please help me with the scenario and have it in one small code?
Thanks
According to answers, I could solve the problem using the codes below in my .htaccess file:
# To remove .php extension
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\ (.*)\.php [NC]
RewriteRule ^ %1 [R=301,L]
# To remove trailing slash
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)/$ $1 [R=301,L]
# To check whether the file exists then set it back internally
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^.*$ $0.php [L]
# To redirect /index to root
RewriteCond %{THE_REQUEST} ^.*/index
RewriteRule ^(.*)index.php$ /$1 [R=301,L]
Now, I got a new problem and that is when I browse the files with a trailing slash, it redirects me to somewhere else and occurs a 404 (page not found) response, like this:
http://domain.tld/page/ => http://domain.tld/page/home/user/public_html/domain.tld/page
Also, if I move the second part to the end or even remove it completely, the server sends me a 500 (internal server error) response.
Finally with a little research, I could find the solution for previous problem using this advice:
https://stackoverflow.com/a/27264788/5420319
So, I've changed one of the codes from this:
# To remove trailing slash
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)/$ $1 [R=301,L]
To this:
# To remove trailing slash
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_URI} /(.*)/$
RewriteRule ^ /%1 [R=301,L]
And I also moved this part to the top regardless of being effective.

Answer 1

According the the answers and my own research, this would be the final answer:
# To remove trailing slash
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_URI} /(.*)/$
RewriteRule ^ /%1 [R=301,L]
# To remove .php extension
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\ (.*)\.php [NC]
RewriteRule ^ %1 [R=301,L]
# To check whether the file exists then set it back internally
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^.*$ $0.php [L]
# To redirect /index to root
RewriteCond %{THE_REQUEST} ^.*/index
RewriteRule ^(.*)index.php$ /$1 [R=301,L]

Answer 2

Redirecting page.php to /page:
RewriteCond %{THE_REQUEST} ^[A-Z]{3,}\ /page\.php [NC]
RewriteRule ^ /page [R=301,L]
Now, setting back internally to page.php:
RewriteRule ^page$ /page.php [NC,L]
Remove trailing slashes:
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)/$ $1 [R=301,L]
Now, checking whether a file for /some/page/foo such that /some/page/foo.php exists:
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^.*$ $0.php [L]

Answer 3

A cleaned up and simplified version of the accepted answer is:
# To remove trailing slash
RewriteCond %{DOCUMENT_ROOT}%{REQUEST_FILENAME} !-d
RewriteRule ^(.*)/$ $1 [R=301,L]
# To remove .php extension
RewriteRule ^(.*)\.php$ $1 [R=301,L]
# To check whether the file exists then set it back internally
RewriteCond %{DOCUMENT_ROOT}%{REQUEST_FILENAME} !-f
RewriteCond %{DOCUMENT_ROOT}%{REQUEST_FILENAME}.php -f
RewriteRule ^.*$ $0.php [L]

Answer 4

Options +FollowSymLinks
RewriteEngine on
AddHandler php5.3-fastcgi php
SetEnv no-gzip dont-vary
RewriteRule ^([a-z]+)$ index.php?page=$1 [QSA,L]
RewriteRule ^([a-z]+)/([a-z0-9]+)$ index.php?page=$1&subpage=$2 [QSA,L]
RewriteRule ^([a-z]+)/([a-z0-9]+)/([a-z0-9]+)$ index.php?page=$1&subpage=$2&module=$3 [QSA,L]
RewriteRule ^([a-z]+)/([a-z0-9]+)/([a-z0-9]+)/([a-z0-9]+)$ index.php?page=$1&subpage=$2&module=$3&submodule=$4 [QSA,L]
You can play with that.