biochemistry - How does a plant grow before photosynthesis is possible?

There are quite a few questions and thoughts in there, I'll try to cover them all:

First, to correct your initial word equation: During photosynthesis, a plant translates CO₂ and water into O₂ and carbon compounds using energy from light (photons).

You are correct to assume the C is further used for the growing process; it is used to make sugars which store energy in their bonds. That energy is then released when required to power other reactions, which is how a plant lives and grows. C is also incorporated into all the organic molecules in the plant.

Plants require several things to live: CO2, light, water and minerals. If any of those things is missing for a sustained period, growth will suffer. Most molecules in a plant require some carbon, which comes originally from CO₂, and also an assortment of other elements which come from the mineral nutrients in the soil. So the plant is completely reliant on minerals.

Most plants, before a leaf is established or roots develop, grow using energy and nutrients stored in the endosperm and cotyledons of the seed. I whipped up a rough diagram below. Cotyledons are primitive leaves inside the seed. The endosperm is a starchy tissue used only for storage of nutrients and energy. The radicle is the juvenile root. The embryo is the baby plant.

When the seed germinates the embryo elongates, the endosperm depletes, the testa ruptures, and the cotyledons emerge from the seed. The cotyledons are green, and like leaves can photosynthesise, so as soon as they are in the light the plant is able to make carbon compounds. The radicle elongates at the same time, and becomes the root, so the plant is very quickly able to obtain fresh nutrients from the soil (or whatever it's growing in).

At all stages of a plant's life it is using both energy stored in carbon compounds (from CO₂) and nutrients which it took up via its roots. At no point does the plant start to depend solely on the CO₂ in the air for its growth.

You are right that the way in which plants acquire energy and nutrients prior to leaves and roots being established varies between plants. Above I outlined the way most plants use. But there are lots of variations. For example, orchids have very tiny seeds, some barely visible to the naked eye, like specks of dust. They have no endosperm or storage tissue, so they have to rely on a symbiotic mycorrhizal fungus to get carbon and nutrients. The fungus grows through the coat of the orchid seed, then provides everything the growing plant needs until it has its own leaves. Then the orchid repays the fungus by providing sugars.

There are lots of other examples, but we could go on all day!