Do you think Dallas and Brownsville are very far away from each other?

Yes, it's a 7.5 to 8 hour drive.

Yes, according to Google Maps, the fastest route takes 8 hours and 2 minutes and 545 miles. In my opinion, that is a really long drive and if you take the fastest route you might encounter traffic in Austin or San Antonio. However, if you fly, it takes a little over 1.5 hours if nonstop.

Not by Texas standards. 

That depends how you define 'very far'. Very far when you're walking is different from very far if you're driving or flying.

I know where they are on the map.  Depends what you mean by very far.  Way further than I would want to walk, but a pretty short plane trip

They're about 500 mi. apart, aren't they?  Brownsville is down at the southern tip of Texas and Dallas is up towards the Northeast.