Is the standard enthalpy of formation always non-positive? If the standard enthalpy of formation is defined as 


  the energy change when 1 mole of a substance is formed from its elements in their standard states


and forming chemical bonds is an exothermic process, then does this mean all standard enthalpies of formation are non-positive? 

Why do some compounds such as $\ce{NO}$, $\ce{NO2}$ have positive standard enthalpies of formation? Is this because to form $\ce{NO}$, we must first break the bonds in $\ce{N2}$ and $\ce{O2}$ (endothermic) to form the bond in $\ce{NO}$ (exothermic, smaller energy produced than the energy required to break the bonds of original standard state elements)?
You'v answered the question yourself.

In general, any enthalpy of reaction is the measure of the energy remainder after breaking the bonds (positive) and what energy we got back from the formation of the bonds (negative). 

The net sum can be positive or negative, and it depends on what the reactants and products were. As you correctly said yourself, $\ce{NO}$ happens to have a positive standard enthalpy of formation because, as experimentally determined, more energy went into breaking the bonds of reactants than what we got back from forming the products.

So,


  Is the standard enthalpy of formation always non-positive?


The answer is no.