Would you consider "Trust & Reputation" as strategic priority? If yes, what drivers can support building a company good reputation?

It depends on the industry.  A basic level of trust and reputation is obviously essential in any business, but if the business is, for example, in finance, then it's more important.

However I would not regard either as these things as a strategic priority, because they are not things that the business can directly control: they are the opinions of other people.  Furthermore, the ways you get them are not really strategic priorities either, they are part of the ethos of the business.  Does the business strive to offer a high quality product with customer service based on honesty and fairness?   Or is it a low-cost, basic quality offering, with customer service at the minimum possible level?  Both are viable business plans, but one will create trust and reputation and one won't. 

If you want to build a good reputation then focus on the things that create that:  quality product, good customer service and - very importantly - consistency.