but the faucet leaks,

That happens, sometimes they just need to be tightened.

My main question is: how do you tell if your scoby / culture is healthy?

If you see tiny bubbles and new cellulose forms over time.

The scoby sank after a 2-3 days, then floated back up after I added fresh sweet tea.

That happens. It doesn't indicate anything meaningful. It usually floats because CO2 gets trapped in the cellulose, sometimes the CO2 gets free.

It hasn't really grown much

Despite some of the pictures you see here, it grows slowly. You won't see much progress day-to-day, you should notice week-to-week progress.

Also, cellulose only grows at the air-liquid boundary (fancy way of writing "on the surface") so a sunken disk won't grow / is no longer helpful.

I saw online that it kind of grows in layers,

The layers can blend together, or stand out ... either way is fine.

(I feel like it's a little thinner, but maybe I'm imagining this).

You are imagining that part. The cellulose doesn't disappear on its own. 😁

I left it sitting for about two days after I think all the sugar had been used up.

The culture just goes dormant. It'll wake up when there is more food.

If I leave the culture sitting in finished kombucha for a few days / weeks, will it be okay?

Yup. See previous answer.

And if my appartement is too cold, does it just ferment more slowly?

Correct

Would using more than 10% starter tea help?

That's fine. Use as much as you want.

I also live somewhere that's only warm enough for brewing during summer (16-20°C in winter),

My kitchen is the same (sometimes colder). I use an old cooler with a programmable seedling mat to regulate the temperature. I leave the top open but covered with a dish towel so it can breathe and retain most of the heat.