No, the Moon doesn't always rise in the East. At higher lattitudes, it can sometimes rise in the North East or even due North in the summer on some days, if you are close to the Arctic circle or beyond it. Where I live in the north of Scotland, then the positions of sunrise / moon rise and sunset / moon set are not that far from each other, towards the North.
It rises approximately in the East year round if you are at low latitudes, and it rises exactly in the East on only two days in the year.
In the Southern hemisphere, the sun and moon can rise in the South.
And if you are at the North or South pole, then the sun describes a spiral path, below the horizon for half the year, and above it for half the year. At the equinoxes it skims the horizon. Making it hard to say where it rises or sets, but then there is no East or West, everywhere is South (or North if you are at the South pole).
If you have mountains as well, or even low hills or undulations in the ice, the sun and moon could rise and set behind them multiple times as it skims around the horizon, especially when they are low in the sky. On those days it could rise and set in any compass direction.
And in winter, at high lattitudes, the sun and moon don't rise at all. And in summer, they never set, at high lattitudes.
There are only two days in the year when the sun and moon rise due East, neither North nor South of East - that's at the spring and autumn equinoxes. Everywhere in the world they rise due East - except for the two poles where the sun skims the horizon (theoretically half of it shows above the horizon if you are on a completely flat ice sheet).
And actually even then it will be slightly North or South of due East depending on your longtitude. For it to rise exactly due East you need to be on the point on the Earth that experiences the spring equinox at the exact moment that the Earth's axis is perpendicular to the line joining the Earth to the Sun. At each equinox, there's only one longtitude that has that property. (But there we are talking about a precision you wouldn't notice by eye :) ).
See also: Find your way using the Sun which has some nice images and animations of the position of the sun depending on the season.