Neptune and Uranus have estimated core temperatures of 7273 K and 5255 K respectively. Uranus and Neptune however likely do not have liquid water oceans. The combination of pressure and temperature on both planet results in unsuitable conditions for liquid water oceans. In fact for liquid water oceans to form on Neptune, it's predicted it would have to cool down.
Paper on this at: Cornell University
(Irrelevant, but fun) Diamonds oceans might exist. Uranus and Neptune are made up of 10% carbon some of which is methane. Methane when exposed to the conditions available in Uranus and Neptune can form diamond. Diamond can melt and form liquid diamond under again, the conditions on Neptune and Uranus. Therefore there could potentially be diamond oceans on Neptune and Uranus.
Paper on this at; Nature Materials
Two plausible theories considered for Neptune source of heat is that as you said, the methane concentration in the atmosphere is high enough to retain the formation heat after the formation of the planet. The other one is that the tidal forces generated by Triton's retrograde degrading elliptical orbit, but the paper on it concludes that it doesn't explain the difference between Neptune and Uranus.
Another thing to consider is that Uranus might be cooler than it should be rather than Neptune being warmer. This could be a result of an Earth sized
indirect collision earlier in its lifetime. The collision causes this decrease in emitted temperature because it deposited a significant amount of material around the core which made heat transfer from the core to the outer parts of the planet much slower. Note that if this collision were to occur, it is what would've caused the unusual tilt of 97.7°.
Paper on this collision at: The American Astronomical Society
To the second question, the methane in the atmosphere of Neptune, as covered before, can produce a significant greenhouse effect.
Neptune emits 2.7 times more energy than it receives from the Sun. This access energy powers the atmosphere to produce the storms that are not seen on its twin planet Uranus. The source of internal energy cannot be due solely to leftover energy from formation (i.e. Jupiter) since Neptune is smaller and would have radiated away the energy long ago. Nor is it due to an unusual chemical change, such as the helium rain for Saturn. Rather, it appears that Neptune is more efficient at trapping leftover formation heat due to the fact that methane is highly abundant in Neptune's atmosphere, and methane is an excellent insulator of heat (i.e. the greenhouse effect). Neptune has a sub-zero type greenhouse effect that is trapping formation heat that should have been radiated billions of years ago like Uranus.
This extract is from:
University of Oregon
In regards to why Uranus atmospheric methane doesn't appear to have as much of an effect as on Neptune, it could be that either Neptune simply has more methane in it's atmosphere or that a combination of factors, some of which mentioned above causes the difference. Beyond that I do not know.
Paper on Tritons Tidal Heating:
University of Texas