Would I be allowed to build my own house?

Yup

If so, would I be allowed to sell it?

Nope, houses would be decommodified. You could move out of it, and someone else who needed it could get it, so they could still be exchanged, but there wouldn't be a market for it. Same way we approach organs, sometimes people don't need organs, and they give organs away, and we have systems for managing organ transplants etc, but we don't have an organ market because the material effects of having an organ market are bad (of course there's still a black market for organs, and there'd be a black market for houses, but it wouldn't be an open market and would be severely less harmful as a result).

The next couple paragraphs are based on the idea that housing is still commodified and being bought/sold, so I won't address those because my above comment already does.

Unequal does not necessarily mean unfair

Correct, but ~10 people having the same wealth as 4 billion people happens to actually be a result of unfairness in the system. We will soon live in a world with trillionaires (estimates are within the next 10-15 years). This means a single person will be able to give a million people a million dollars worth of capital. That's a symptom of a failed system, not a cause of it. It's not unfair because it's unequal, but it is unequal because it's unfair. Capital shouldn't be a method by which you can accumulate more capital. It allows infinite feedback loops of extraction that end up materially hurting the vast majority of humans. Wage stagnation is a real thing that has happened for the last 40 years as we eased restrictions on capital. Stock buybacks for example were illegal from 1920->1980, now stock buybacks are used more than ever and allow massive amounts of capital accumulation for corporations without reinvestment of capital into materially beneficial avenues like R&D. There are many ways unrestricted capital allows for exploitation and materially detrimental outcomes.