Learn Portuguese (Portugal): O próprio professor levanta a barra com melhor postura. - The teacher himself lifts the barbell with better posture.

When placed before the noun, próprio acts as an emphatic word meaning "himself" (as in, "the teacher himself"). If you put it after the noun (o professor próprio), it changes meaning to "the suitable teacher" or "the proper teacher".

Could I use mesmo here instead of próprio to mean "himself"?

Only if you change its position! If you put mesmo before the noun (o mesmo professor), it means "the same teacher". To use mesmo for emphasis ("the teacher himself"), you would have to put it after the noun (o professor mesmo). However, o próprio professor is much more common for this.

Why is there no article before melhor? Does it mean "better" or "best" here?

It means "better" here. In Portuguese, melhor is used for both "better" and "best". The definite article is what tells them apart. Com melhor postura means "with better posture", while com a melhor postura would mean "with the best posture".

Does barra always refer to gym equipment?