I know that the universe:
- It's around 13.772 billion years old
- It expands
But it's not clear to me if this is not merely the age of the farthest known photons which reached Earth.
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Possibly you are making the mistake of thinking of the universe as expanding from some central explosion? That is not so. All parts of the universe were part of the big bang.
The "oldest" light we can see was emitted about 13.7 billion light years ago, approximately 400,000 years after the big bang. That light has travelled 13.7 billion light years to get to us. But there are very old stars that are much closer to us. Some of the stars even in the solar neighborhood are more than 10 billion years old. The light emitted just after they were born has now travelled 10 billion light years into the cosmos.
When we look out into the universe, we see back in time. But the big bang occurred in every direction that we look.
After the Big Bang the universe was full of hot plasma, which both absorbed and emitted photons. The result was that the oldest photons we can see come from the end of that era (a few hundred thousand years after the big bang); they make up the cosmic microwave background today.
The age of the universe is calculated by fitting observed data to a cosmological model and calculating how long ago the scale factor (the "radius" of the universe) was zero. This is how you get the 13.772 billion year number - this comes from by using the $\Lambda$CDM model with the best parameters we know today. As data and theory gets more refined the number will be updated.