I'm new to quilting and adding a border that needs to run in a specific direction. My question is, can fabric be torn/ripped in both directions, selvage to selvage and parallel to the selvage??

Yes, however there will be a little bit of damage to the fibers of the fabric. I will rip for my backings but then use a 5/8" wide seam to cover the damage area.

I suppose you could rip a little bit wider then trim off the damaged part. With a 1/4" seam, you may not want to rip and just sew.

I do rip off smaller pieces off of yardage to cut for projects. For instance if I have a 10 yard piece of fabric and I need only a 1/4 yard piece to use, I will frequently rip off just over what I need and then press and cut my pieces, but at least 1/4 to 1/2 inch away from the ripped side.

Yes. In fact, I've found that tearing strips parallel to the selvage yields wonderfully straight borders.

I rip to make a straight edge then trim off the fray. It speeds the processes.

I rip fabric both on the cross grain and lengthwise grain frequently. I increase the measurement a bit and trim the edge were it was torn. It is my preferred method of dealing with long cuts for borders and for finding the true grain line on the cross grain.

I am old enough to remember when all my sewing projects began with fabric nipped by the measuring device and then ripped across the width. Older than dirt?

Yeah, Cheshire, me too.

This is discouraged by some people as they dislike the waste of fabric as you trim off the stretched fibres on the rip.
I must admit to ripping my backing fabric to the required length.
I have never thought about the direction as both work and are useful tool as long as you remember the stretched fibres at the edge

I have been ripping for more than 40 years. It is the only way to get the true grain of the fabric. Having the true grain allows you to make the best finished project. The only other way to get the true grain is to pull a thread and then cut along the space it leaves. Pulling a thread is not easy and can be very frustrating. Working with the "true grain" makes a finished project that looks and hangs professionally. In making a garment it means the difference between looking "homemade" and professional.