سلام سعید جملهت خلی خوبه "Founded" means to have established or originated something - but it usually refers to an institution or organization. So it has a slightly narrower meaning than "establish". A laboratory is not an institution or organisation, so "established" would be better in this context. The following additional small change would also make the sentence a bit more formal - "The public health laboratory has been established to satisfy the medical needs of the local people."

If you are saying the new laboratory has been founded to satisfy ... (as in built or established to satisfy) then yes, all good as written. If you are saying the laboratory has been found to satisfy (as in: other people agree that it does what it was meant to do) ... then found is ok. Establish can also be used in this sense too but not in that construct. It might work like this: It was established that the laboratory satisfies the local people's needs. 'The hotel was founded' is an unlikely start to a sentence but it might be right in some situations. I tend to think of only using founded for important things or groups.

110% correct - yes. Reads well and wouldn't attract any further scrutiny if written in an English test.