the reason the date isn't being used in the first type='date' field is because that's already a php string. don't use <?= ?> tags around the variable, just use the $newdate variable - <input type='date' name='date' value='$newdate' >
next, nested forms are invalid. the date search form needs to be closed with a </form> tag, before you start the next form.
several of the form fields have the same name, so, only the value from the last one will be used.
the post method form processing code should be on the same page as the form. this will simplify all the code and allow you to repopulate the field values if there is a user/validation error in the form processing code.
all those lines of code copying one variable to another is a waste of your time typing. just use the original variables.
why on earth are you using the PDO database extension to get the existing data to be edited, then using the mysqli database extension in the post method form processing code? just use the much simpler PDO extension everywhere. also, use a prepared query for the UPDATE query and the UPDATE query needs a WHERE clause so that you are updating the correct row.
lastly, if there can be more than one row per date, you need to loop to fetch and produce the edit form(s) with either one total form or one form per row and you would need to use an id (autoincrement primary index) to determine which row to update.