If I call someone while signed onto Skype, will my phone number show up on their end?

As opposed to physically dialing a phone number, I used my iPhone to call someone on Skype after signing on to Skype with my e-mail and password. Therefore, I simply clicked the "phone' icon. Since I was using Skype from my iPhone, will my mobile number show up in that person's missed calls?

Assuming it is a Skype to Skype call (VoIP) then no. It would use your Skype account (username) rather than a phone number.

Skype does have phone call abilities that is pay for last time I checked.