Relationship between Lewis Theory and Redox theory?
Under the Usanovich definition of acids and bases, any substance which loses electrons is a base while any substance which gains electrons is an acid. Here, any redox reaction is also an acid-base reaction, whereas Lewis theory has nothing to state in these cases. 

Regarding your second question, blanket statements are dangerous. Depending on exactly what you mean, I can provide a counter-example. Here is a strong oxidant ($\ce{HNO3}$) reacting as a Lewis base:

$$\ce{HNO3(l) + H2SO4(l) -&gt; H2NO3^{+} + HSO4^{-}}$$