As far as age is concerned it's an almost certainty
Meah. We just figured out (about ~100 years ago) that there's such a thing as other galaxies and that the universe seems to be expanding. Today we're strongly considering the possibility that we don't know almost anything about dark matter & dark energy which seem to make up > 90% of everything there is in the universe. We have 2 major theories about how physics works at the grand and small scale and we know for sure both are insufficient and will need major corrections before we can claim we have one coherent theory.
My point is that we're slowly discovering just how ridiculously little we know about the universe so we don't have the luxury of considering anything almost-certain when it comes to its age or size.
As far as volume is concerned ... How do they come to such a suspicion without any smidgen of evidence?
I'm afraid there is no evidence obtainable about the volume so I would err on the side of caution and consider the volume infinite. Here are 2 interesting issues to consider on this topic:
If the universe popped into existence through the Big Bang then it seems that's when spacetime was created and so rudimentary logic would tend to support the idea that spacetime is finite. Better not ask questions like "spacetime was all created at once or is it still being created as the universe expands?" or "spacetime expands into what?".
When we look around, we get information from the farthest corners of the universe. Some of it has been traveling towards us ~since we think the Big Bang occurred and only now it was able to reach us. It would be tempting to say this distant information came at us from the edge of the universe and so the universe is finite. Unfortunately, it's not that simple. In an expanding universe, we know for sure some distant information will not be able to reach us because the universe expands and the information can't travel towards us fast enough to compensate (speed of light limit). So it's very possible there's lots more information (i.e. universe) beyond the information horizon we're able to experience, that we'll never be able to see because of the expanding universe. So the universe could very well be infinite and it could be physically impossible for us to know anything about it beyond a certain threshold (I find extraordinary the irony of discovering we're physically limited from knowing anything about the universe both beyond a certain grand scale and beyond a small scale).