Hi, and good question. There are quite a few good threads on here about polarity if you do a search, but I am new to the forum as well and understand the search function can be something we overlook.
To answer your question it would help if we knew what your current understanding about polarity is and maybe we could help correct any incorrect ideas you may have. Also, it would help if we knew the specific molecules you are looking at and we could take a big picture approach to your question. It is easy to memorize the idea that, for example, an alcohol is polar, but if we look at the example of tyrosine many places have it listed as non polar despite the -OH. In cases such as this I have come to think about it as the non polar groups outweighing the polar groups (some one can probably provide a more scientific way of thinking about it, that would be helpful). The idea that an amide isn't polar I'm not sure how to answer that. Polarity will generally boil down to an uneven sharing of electrons AND a molecular shape that causes a dipole moment. A good example would the difference in CO2 and dimethyl ether. CO2: O=C=O, while the oxygen is more electronegative than the carbon, the dipoles will cancel each other out. This leaves CO2 non polar. Dimethyl ether: CH3-O-CH3 (with 2 lone pairs on the oxygen). The oxygen is sp3 hybridized which would generally be a tetrahedral geometry, however because of the 2 lone pairs the geometry is "bent". Because of the bent shape the molecule is polar, parts of each dipole moment will add together to form a net dipole and thus a polar molecule.
For the hydrogen bonding, the answer to your question is yes. The lone pairs on those elements can hydrogen bond with your suggested H's. Once again, though, it is important to consider the big picture, and you seem to have thought big picture with your specific question about ether. The lone pairs of the the O in an ether can hydrogen bond with appropriate H's, but those hydrogens must be present. In a pure ether solvent there will be NO hydrogen bonding because there is no H bond donor, only the acceptor ether. It has been helpful for me to read each MCAT question with the idea that it is an individual unique question with individual unique parameters. So, in the case of your specific question that stated an ether cannot hydrogen bond it may be helpful to instead think "within the parameters of this specific and unique question, the ether cannon hydrogen bond". That should help get you on to the right track of thinking. If the question was not talking about ether hydrogen bonding with itself or some other non H bond donor then perhaps there is something else going on and some one else can provide some more guidance (perhaps in that specific question sterics were not allowing hydrogen bonding? I'm not sure...)
Hope there was something helpful in there!
