If the entire universe is the same age, can one assume that any potential life elsewhere is at the same stage of development as us?

At least part of this question is more about biology than physics, so let's first cover the physics:

In the standard cosmological model ($\Lambda CDM$) the big bang occurred everywhere in the universe 13.7 billion years ago. So, yes, all parts of the universe are the same age. However when you look out to great distances you are actually seeing back into time since the light has to travel that distance. For example, the images you can see of the Cosmic Microwave Background radiation was emitted when the universe was only about 380,000 years old.

The matter content of the universe just after the big bang would have only had hydrogen and helium and therefore life (as we know it) would not have been possible since significant amounts of carbon and oxygen (and other elements) are needed. These heavier elements are created inside of stars and when they explode in a supernova the heavier elements are dispersed into the interstellar medium from which other stars (such as our sun) can form. The first stars probably were much heavier than our sun and would have made lots of heavy elements and exploded in a supernova in significantly less than 1 billion years. (For more information about the first stars (Pop III) see this question.)

However, our sun is only about 4.5 billion years old so the sun formed when the universe was already 9.2 billion years old. So even if it took a couple of billion years for the first stars to produce enough heavy elements and for stars with a mass near one solar mass to form, there still would have been a possible 7+ billion years for stars like our sun to have formed and for life to have evolved to levels far beyond our level of civilization.

The biggest uncertainty is in the biology. We only have 1 example of the evolution of life so we don't have good statistics. The good news is that the first life seemed to have been formed within a few hundred million years of the earth's formation 4.5 billion years ago. So it could be argued that life forms easily and quickly. On the other hand life stayed in a very simple mostly single cell form for the first 4 billion years and only recently (last few hundred million years) developed into complex forms. So there is probably a lot of variation in how long life takes to develop into complex forms - but that is pure speculation since we only have a sample of N=1 to deal with.

So in general, no, we cannot assume all life in the universe is at a similar stage of development as ourselves.

Yes, if the Universe is very uniform, life development is slow and linear.

Otherwise, no.

Given that the history of life on Earth is quite non-linear, I guess the overall answer is no.

Possible but not necessary and highly unlikely.

Reasons:

Not necessary part: Rate of development of life on one planet may differ from that of another planet.

Unlikely part: Town that is 10km from my place is in different stage of development than my own village. We see different civilizations, different countries on this planet took so different paths in the history and are at different stages of developments.

Suppose, we build planet that is exactly like earth at the time when evolution began. And simulate evolution using it. And wait till "2012" will be in the same stage of development?