Custom forms + Polylang

I am new to Wordpress but I have some knowledge of PHP. I am creating a bilingual site (with Polylang) which has many forms. Some of them are handled by Contact Form 7 but some others have to be custom made.

So, the first form I made was quite a disaster - it works but under certain circumstances.

In the archives page of a custom post I insert

form method="post" action="?php echo esc_url( home_url( '/' ) ); ?"
input type="hidden" name="post_type" value="myvalue"
select class="form-control"  name="k" id="k"
?php  
for($i=0;$i2;$i++){?
option value="?php echo $array3[$i]?"?php echo $array3[$i];?/option
?php }?   
/select
span class="input-group-btn"
button class="btn btn-search" type="submit"i class="fa fa-search"  /i/button
/span
/form 

I created a template file named mysearchresults.php In there I inserted the essential (between them)

 echo $_POST['k'];
 echo "Hello world";   

I created a page having as template mysearchresults.php in one language (named mysearchresults) and another in the other language (same template) named mysearchresults-2.

Loading the two pages gives a page with the Hello world printed.

Adding

form method="post" action="?php echo esc_url( home_url( '/' ) ); ?mysearchresults-2"

gives 404 (when I press the submit button - having made a selsction) - the same as instead of -2 I put mysearchresults. Putting mysearchresults.php gives the same error.

So I tried another approach, instead of

select class="form-control"  name="k" id="k" 

I put

select class="form-control"  name="s" id="s"

and for action I left

action="?php echo esc_url( home_url( '/' ) ); ?"  

In the search page I put echo $_POST['s'] and it loads the search results page printing the value of my selection.

What am I doing wrong? How can I have a custom form, pass the values in my action file and make operations on them? Why leaving like this the action label it loads the search results page?