Am I considered a full-time student if I take online classes through my university?

You need to be enrolled at least half-time to qualify for an education credit, which I surmise you are concerned about. It wouldn't matter if you were just taking online courses.

Per IRS regulations ""You have to be enrolled at least half time in a program leading toward a degree, certificate or other recognized educational credential for at least one academic period during the tax year."

You can read more about that here:

https://www.irs.gov/credits-deductions/individuals/education-credits-questions-and-answers

As a practical matter, it will often by marked on your form 1098-T if the educational institution considered you to be enrolled at least half-time.

Yes, online counts as class attendance.  You are full time if you meet your school's definition of full time.