Is it okay to use a different mouthwash name brand and toothpaste together?

Certainly: The two are independent of each other and will not interfere with each other in any way.

Mouth rinse: No issues if you mix any brand of tooth paste with any commercial mouth wash .No fear of chemical reactions.

Yes: Mouthwash and toothpaste are not paired. However, i personally do not feel that it is usually necessary to use a mouthwash routinely unless recommended by your dentist or doctor.

Mouthwash Overrated: Ever wash your car with a hose & let it dry? How clean does it get? You have to mechanically rub the surface to really get it clean. Teeth are the same. Where ever you can't reach & clean with floss or a pick & toothbrush stays dirty! use mouthwash first to reduce the bacterial count, then floss or pick & lastly brush with toothpaste. Most importantly, do not rinse so the Fluoride can soak in.

Mouthwash: I do not think there is anything in otc brands that would be specific or interact. Mouthwash will eliminate some bacteria, temporarily as well as freshen breath but will never replace brushing and flossing. Some perscription mouthwash do more as well as ones for dry mouth or increased need for flouride, but they are an adjunct to proper oral hygiene.