Skype is no longer supported on Windows Phone 8.1 as of July 2017. Your options pretty much boil down to the following:
Upgrade your phone to Windows 10 Mobile (the Lumia 636 is officially supported; download the Upgrade Advisor app to verify eligibility and get started)
Buy a newer Windows phone running Windows 10 Mobile
Use another instant messaging app (...
I notice that apps like whatsapp switch from loudspeaker to earpiece speaker if you hold the phone to your ear. It was so simple, I thought I had a weird bug where it would switch between the two, then I realized it works if I just hold it to my head. Perhaps this is the same for Skype or for the media player in general on the phone?
To sign in to Skype with your existing Skype account, you'll need to link your current Skype account with your Microsoft account. There’s plenty to gain when you sign in to Skype with your Microsoft account:
one less password to remember
improved account security
even easier and more secure password recovery
access to the Microsoft online services
To link ...
I haven't done this, but if you link your Microsoft and Facebook accounts I'm pretty sure this is possible, for friends that have both Skype and Facebook accounts.
You can definitely open someone's page in the Contacts application and Skype them from there. I've also switched to a Skype video call from a regular phone call as well.
You can't. Although the Skype Preview app is able to use legacy (voice+SMS) part of your cellular connection* to send SMS, it does not support cellular voice calls. All Skype calls (voice or video) are over the Internet, using either your WiFi or your cellular data connection. You need to continue using the Phone app to place calls using your usual minutes (...
You can disable the skype integration in the messaging app and use the standalone Skype Application (to not mess around with your text messages, for your n°1 problem), in which you can choose to appear online or, as said above by @Aventinus, invisible. (for your N°2 problem).
To avoid using the built in Skype tools sign out by going to Settings inside of ...
Skype integration has always been disappointing in every mobile OS. Unfortunately the only option in terms of status is choosing between Online and Invisible. Switching to Invisible may save you from some of the spam.
However, there should be a Sign Out option. (I have currently migrated back to 8.1 so I can't check).
There's a final solution to this ...
I think this is a headset limitation, I use creative headsets and it works fine I can make calls, answer call and even redial the last number or Skype contact I had a call with.
And all calls go through my headsets nothing rings on my phone.
If you can try another headsets from a different brand and check if the problem persist
Windows Phone OS like many of the popular and relevant phone operating systems of today are designed with the concept of "always connected" and "connected across all devices" in mind. This results in the situations like the one that you are experiencing.
You cannot trigger a particular action on your phone (at least not at this time) that would happen when ...