If matter comes from energy, does this mean that energy has weight?

What would energy having "weight" mean?

Relativity teaches us that $E = \sqrt{(m_0 c^2)^2 + (p c)^2}$ where $m_0$ is rest mass, $c$ is speed of light and $p$ is momentum. This is the mass-energy equivalent you are referring to, and it allows particles without rest-mass (e.g. photons) to convert into particles with rest-mass (e.g. electrons) by "investing" extra energy into "generating" a rest mass.

We can consider having "weight" as having interaction with a gravitational field. Then we can rephrase your question in a more well-defined way: Can particles without rest mass interact with gravitational fields? The answer is yes, and this can be seen for example in gravitational lenses, where the path of light is bent by the gravity of a massive object (e.g. a star). A more dramatic example is a black hole, which is so massive that light cannot escape its gravity.

Be careful, though. What we in our daily lives call weight refers to the rest mass, which is constant. So in that sense no. For example, the rest mass of a football does not change if you invest energy into it by kicking it.

For your second question: The total energy of the universe is constant even under inflation (to the best of our current knowledge). The "weight" of the universe is not a well-defined concept, since there is no outside gravitational field to interact with.