Nucleophilicty and Basicity of OH and F
Why is $\ce{OH-}$ more basic than $\ce{F-}$? 


To answer this question, let us consider the $\mathrm{p}K_\mathrm{a}$'s of the conjugate acids of these anions: $\ce{H2O}$ and $\ce{HF}$. $$\mathrm{p}K_\mathrm{a}(\ce{H2O})=15.7\ \\  \mathrm{p}K_\mathrm{a}(\ce{HF})=3.17$$

You didn't need me to tell you that $\ce{HF}$ is a stronger acid than water though, but it is -- $10^{12.53}$ times stronger to be exact. This means that this is also how many times stronger $\ce{OH-}$ is than $\ce{F-}$


  In a polar aprotic solution, why is $\ce{F-}$ a stronger nucleophile than $\ce{I-}$?


Small, electron-dense ions are always great nucleophiles, except when the solvent crowds the nucleophile with hydrogen bonds, as in the case of $\ce{F-}$. It is so heavily surrounded by solvent molecules in polar protic solvents that it can't make an attack very well. $\ce{I-}$ is not as heavily solvated in these solvents, and it is very easily polarizable, making it a better nucleophile in these solvents.

In polar aprotic solvents, the small $\ce{F-}$ ion can easily slip between solvent molecules and make attacks. Now in comparison, $\ce{I-}$ is a much poorer nucleophile.


  In a lecture, my teacher said $\ce{RO-}$ is relatively stable because of the high electronegativity of $\ce{O}$.


Again, we can answer this by looking at $\mathrm{p}K_\mathrm{a}$'s. The $\mathrm{p}K_\mathrm{a}$ of ethanol, for example, is $15.9$! Only $10^{0.2}=1.58$ times weaker an acid than water! That means that ethoxide will deprotonate everything $\ce{OH-}$ will and then a little more, so it's not really that much more unstable.


  When people say something is stable, are they referring to its acid/base characteristics, or can stability also be referring to nucleophilicity/electrophilicity?


The two are pretty well related. You have exceptions like how $\ce{F-}$ is a poor nucleophile in water despite being a good base, but as a rule of thumb, I would say people mean acid/base reactivity when they talk about "stability."