Is the net angular momentum of the Universe zero?
Is the net energy of the Universe zero?
The net angular momentum of the Universe, according to most of the observations that have been made so far is indeed zero. A spinning universe has some interesting effects. Among the more bizarre is a solution to time-travel that allows for closed time-like paths. There are relativistic effects that theoretically should be visible for rotating universes. Such effects have not been observed.
Net energy of the universe is a rather difficult term to describe. Energy is best measured as a difference between two states. As the universe itself is a closed entity, it’s rather difficult to describe what it’s “net energy” is. Most of the time, what we talk about is the energy density of the universe. This density has numerous components due to mass, radiation, dark energy, and curvature. It looks like the densities of all these components combine right now add up such that gravity in its GR formulation is exactly balanced by the expansion. This is the so-called “flat” universe. Theoretically, the energy density of the universe is equal or EXTREMELY close to this critical density. However, because GR is weird it does not mean that the universe is preferentially expanding at a rate that it is ever slowing down toward some equilibrium point at time equal to infinity as you might expect with normal gravity. Rather, the dark energy that is posited to exist due to present supernovae observations provides a repulsive effect in GR that looks like it is ultimately driving our universe to the heat death end and a very empty and lonely place indeed.
1: Probably. If it’s not, then we’re in a Gödel universe, in which closed timelike curves exist. Time travel, in other words. And most folks think that time travel is probably impossible.
2: Depending on how you look at it, either not defined or probably yes. You can consistently put the zero point of energy anywhere, for almost all purposes. You might not like the idea of negative energy, in which case you set your zero at whatever the lowest possible energy is for whatever system you’re looking at. But if you don’t mind zero energy, then the math all works out the same.
On the other hand: Some ways of defining energy are simpler than others. If you define gravitational potential energy in the simplest possible way, then it works out to always be negative. It seems quite probable that the negative gravitational energy of this sort exactly balances all of the various positive contributions to the energy of the Universe.