**Incorrect assumptions**
>[OP] we see that entropy can be transferred between a system and its surroundings, with $\Delta S_\mathrm{system}=-\Delta S_\mathrm{surroundings}$

This equation is usually not correct, except when you have a reversible process (an ideal situation where something happens even though everything is at equilibrium). For an equilibrium, you are right: The entropy does not increase.

>[OP] If entropy _is_ a form of energy, then how can universal entropy tend to increase?

Entropy is not a form of energy. It does not even have the same dimensions. Also, there are forms of energy that increase without breaking the first law. You could have a space heater turning electrical energy into thermal energy. The first law can't be applied separately to electrical energy ("the electrical energy in the universe is constant" is not true).

> [OP] If entropy is not a form of energy, then how can it be compared with actual forms of energy and measured in energy units?

It is not measured in energy units. The term $T \Delta S$ is measured in energy units. Consider speed vs time and speed vs distance. They share units, but that does not mean speed is the same thing as time, or as distance. And laws about distances or time do not automatically apply to speed.

**Simple counter example**

If two bodies of different temperature are brought into thermal contact, they will reach thermal equilibrium (same temperature). The thermal energy lost by the hotter body is equal to the thermal energy gained by the colder body (first law). The entropy lost by the hotter body is less than the entropy gained by the colder body (entropy increases, second law).

**Why there is no contradiction**

Lots of quantities have the same dimensions as energy (work, heat), and there are a lot of forms of energy. The first law does not apply to any of those, just to all energy combined. So applying the first law directly to entropy does not make any sense, and entropy is not a form of energy.