Civilization – What determines how quickly a ship repairs

I'm assuming you mean shipyards, not starbases. Or this is a starbase feature I've not yet discovered.

If you go to the shipyard, select it, and click "Manage" (or doubleclick on it), there's a "Edit sponsors" button on the bottom left.

Below the name of the planet, there's some numbers. I forget what they all are, but the second-to-last number indicates the tile distance between the planet and the shipyard (which is relevant as production decays more over longer distances), and the last number indicates the resulting production the planet would add to the shipyard.

This is the result in production, so the number will be affected by you changing things like production focus or social/military production division on the planet.

Edit

From memory, the line of numbers is something like:

(icon) 3.0 / 6.0 (icon) 15 (icon) 2.6

In order:

3.0 / 6.0 means the planet is generating 6 production, with a 50% split to military production (meaning 3 of its production goes to military projects)

15 is the tile distance to the shipyard.

Given how the shipyard is 15 tiles removed from the planet, the 3 production it sends will be cut down to 2.6 because of the distance decay.

All buildings provide some kind of bonus to the level of adjacent buildings of a certain type.

Factories provide a bonus to the level of adjacent "manufacturing" buildings, Markets provide a bonus to the level of adjacent "wealth" buildings, and Embassies provide a bonus to the level of adjacent "wealth" "tourism" and "influence" buildings.

In this instance it provides a bonus to the level of adjacent "population" class buildings (such as farms), rather than to population itself.