I want to explain some idioms from the question and from other answers, to give you a chance to understand and compare them:

• streng and eng
  »and« is not a German word, but even »streng und eng« will not work in German. Nobody will understand what you mean. Maybe some English native speakers might understand it because they might translate it back into english, and then understand the English idiom.

»streng and/und eng« is like translating German idioms word by word into English, like in this example:
»My English is not the yellow from the egg, but it goes«. (The German idiom is: »Mein Englisch ist nicht das Gelbe vom Ei, aber es geht«, and a correct translation would be: »My English is not exactly brilliant, but it works.«)

• streng
  This word can mean: severe, strict, stiff, stringent, draconic, grim, austere
  None of this meanings has a really positive connotation. »Streng« is a property, that police officers, judges, teachers, parents and directors can have.
• eng
  This word has a very different meaning. It can mean: tight, narrow, close
  If an alley is so narrow, that hardly one car can pass through, then it is »eng«. Trousers, that are so tight, that you hardly can close the zip, are »eng«. In German there is absolutely no idiomatic connection between »eng« and »straight and narrow«. (See also »engstirnig«)
• engstirnig
  When I did read »streng und eng«, then my first association was »engstirnig«, because »eng« doesn't make any sense in combination with »streng« (as described above). But »engstirng« is: narrow-minded, small-minded, parochial, blinkered, hidebound, close-minded
• sittenstreng
  This word carries all connotations that are in »streng« (see above), plus all connotations of »Sitte« or »sittlich«. The noun »Sitte« can mean morals, convention, customs, manners
  The adjective »sittlich« is: ethical, moral
  So, »sittenstreng« is puritanical, strait-laced, strict and grim with morals.
• prinzipientreu
  The noun »Prinzip« means »principle, set of believes« and the adjective treu is faithful, allegiant, loyal, constant. So the compound adjective prinzipientreu is principled, faithful to the own convictions, but it also has a connotation of »... without thinking about wether this convictions are right or wrong«
• tugendhaft
  There is not really a good englisch translation for the German noun Tugend. I just find virtue and goodness, but both have much a much wider field of meaning than the German word. A »Tugend« is a morally high-quality property, that a person can have. And the adjective »tugendhaft« is used to describe the morally flawless behavior of such a person. It has a strong connotation with ethics and moral. I think morally is a good translation for tugendhaft.
• auf dem (rechten) Pfad der Tugend gehen
  This is a idiom that just is a longer way to say »tugendhaft« or »sehr tugendhaft« (very morally)
• tadellos
  The noun »Tadel« means blame, flaw and the suffix -los is -less. So this adjective means blameless, flawless, immaculate and has a respectful positive (even a little bit admiring) connotation.
• frei von jedem Tadel
  Just a longer way to say tadellos
• gesetzestreu
  I explained treu already (look at prinzipientreu). The noun »Gesetz« is »the law, the laws. So gesetzestreu is law-abiding.
• anständig
  The adjective anständig is derived from the noun »der Anstand« which means decency, decorum, integrity. So anständig is fair, decent, modest.
• hochanständig
  The adjective hochanständig ist just an enhancement of anständig.
• pedantisch
  This is pettifogging, finical, micrological, niggling
• geradlinig
  Literally this means along a straight line (gerade is straight, direct; Linie is line, route) and can be translated as upright, straightforward, no-nonsense, straight
• gewissenhaft
  This is conscientious, scrupulous, assiduously, painstaking
• solide
  firm, respectable, strong