Learn English – Going well vs. going swell

In the second paragraph ("I want to see the world, and that I will!") the meaning is: "I am very determined to do much travel and see the world." The "and that I will!" fragment could be re-written as "and I am very certain that I can ensure that this will occur through my efforts!"

In the last paragraph ("who can appreciate sarcasm and dish it right back") the meaning is: "I like the use of sarcasm and I use it myself, particularly when directed at me."

There is no doubt in my mind that he meant "I'm free", or that he had time to talk.

Here are some relevant definitions I found.

clear

6. a : to free from obligation or encumbrance

clear
not blocked or filled; open or available:
The only time the doctor has clear today is 3:30—can you make it then?

However, from personal experience, I don't feel like "I'm clear" is common or popular. I feel like clear is usually used in the following sense

I've cleared my schedule.

I think "I'm free" is much more common than "I'm clear". Regardless, he did mean that he has time to talk because he was using the definition(s) given above.