Detail 1: I have a login form with a login button but there is no button to register.
Solution : Create a login page and in text mode, paste the following code that you adapt according to your need:

[frm-login remember="0" redirect="www.mysite.com/mypage"]
<p style="text-align: center;"><a title="Register" href="www.mysite.com/register"><input type="submit" value="REGISTER" style="height:40px; width:100%; -webkit-border-radius:4px; font-size:15px; background:#ed811c" /></a></p>

Detail 2: I have a registration form and I want to differentiate triggered notifications when creating and editing.
Solution : A hidden User ID field is blank when creating, so just use the conditional test in the notification. Here is an example of a notification message (Replace 1228 with the ID of your User ID hidden field:

<p>[if 1288]Hello[/if 1288][if 1288 equals=""]Welcome[/if 1288],</p>

[if 1288]
<p>We inform you that the modification of your personal information has been taken into account.</p>
[/if 1288]
[if 1288 equals=""]
<p>You have just created your account on [sitename] and we congratulate you.</p>
[/if 1288]

Detail 3: After an update, I want it to appear a message and that an automatic redirection to an URL is triggered.
Solution : In settings forms --> General --> Message, put in the "Updates" field:

Your data has been updated.
<script>
window.open('https://www.mysite.com/mypage', '_parent');
</script>

Here are some suggestions, there may be other ways to achieve your goals.