How does a budget deficit help grow a trade deficit? What are the mechanism that explains this, and does this help explain the U.S.-China trade deficits?
It all boils down to the macroeconomic identity

$$ (G -T) \equiv (S – I) +(M-X), $$

where $X$ stands for exports, $M$ for imports, $G$ is government spending, $T$ are taxes, $S$ private savings and $I$ private investment.

Given $S-I$, the larger is the budget déficit, the larger the trade deficit is. $S-I$ can be thought of as the private sector excess financing ability. With this amount, it needs to finance both the budget deficit and pay for the difference between what it is able to sell abroad and what it buys abroad. If, as in the US, $S-I$ is not too large (due to low $S$), as soon as the government incurs in a budget deficit, a trade deficit will appear.

This does not tell us with which specific country the trade deficit will happen. However, it does tell us that those countries with high $S-I$ and low budget deficits (or even surpluses) need to be running trade surpluses, and this is the case of China (or Germany).