Hello,
1)Hydrogen bonding only really occurs when F-H, N-H and O-H bonds are present. This is because there is such a large electronegativity difference that the hydrogen is pretty much "naked" and has not very much electron density shielding its partial positive charge.
Because the electronegativity difference between C and H is very small (about .35), the sharing of electrons is pretty much equal and you don't get this polarization. As a result, CH4 cannot hydrogen bond.
2) Induced dipole - induced dipole is another name for london dispersion forces - as each molecule's electron density is a probability function, there can always be an instant where the electrons are unevenly distributed and the molecule can be partially positive at one end, and partially negative on the other end. This influences a molecule right beside it, and this propagation results in the attraction between particles (London dispersion forces). As you have also pointed out, you can get at this by elimination.
Please let me know if this is unclear.