Would somebody born 'reversed' (e.g. heart on their right side, etc.) be completely healthy?
Yes, such a conditions exists. It is called "situs inversus" (literally "inverted sites"). All major organs are placed on the opposite site, the condition is relatively rare with an incidence rate of 1:10.000. There are some even rarer forms occuring, when the organs are flipped, but the heart is still on the left side (levocardia) as opposed to the case when it is also on the other side (dextrocardia). 

While some  of the patients are phenotypically normal (besides having their organs on the "wrong" side), congenital heart disease and ciliary disorders (also known as Kartagener syndrome) can occur. See the linked paper below for an overview.



For more information see here and here and the article:


Situs inversus and ciliary abnormalities: 20 years later, what is the
connection?