The Universe is believed to be 13.7 billion years old, but have all those years been the same length

We consider a "year" as 365.25 days long, right?

So have all those 13.7 billion years been 365.25 days long? or 31,557,600 seconds long?

good question. the term 'year' has many meanings. when using the term within a scientific context (the julian year) it is generally regarded to be 356.25 days. however this does not describe the actual length of any particular year within that time. so the statement that something was so many years ago does not imply that all the years are of the same length.

Yep same years. That's because we, the System Internationale, define those intervals so they are standardized. You may note that once every so many years, the SI adds a second or two to the length of a day. That's because old tired Earth's rotation is actually slowing down. It takes longer to go once around its own axis, the day. So the year definition has to change as well to keep it the same interval in terms of days.

From recent discoveries in the Southern Hemisphere, time appears not the same back then as it is today, the speed of light sppears not the same back then as it is today. I think that they are finding that time moved faster back then as well as the speed of light. Traditional theoretical Physicists are wringing their hands as I write.