It's said that at the moment of the Big Bang, space expanded so that to an objective observer matter moved at thousands of times the speed of light. If so, how can we say the universe is 10-30 billion years old? Couldn't it be a billion years old with objects 10-30 billion light years away because of "riding the space wave"?
We estimate the current age of the universe by measuring the expansion of space and extrapolating backwards, taking into account the matter and energy density in the very early universe.
The universe is about 13.8 billion years old, and the observable universe has a diameter of roughly 93 billion lightyears.
The whole universe is likely infinite in size, and if so, it's always been infinite for all time $t>0$, where $t=0$ is the instant of the Big Bang; the size of the universe at $t=0$ is not mathematically well-defined. Please see Did the Big Bang happen at a point? for further details.
And even if the whole universe isn't infinite then current measurements of its global curvature indicate that it's most likely many times larger than the observable universe.