Definitely worth checking for medical causes. Water infections and ear infections have few outward symptoms, just pain. Also the obvious one - teething. You are right in the middle of cutting molars territory.
It sounds like your DD doesn't handle light sleeping very well. When you get the timing right at bedtime, she probably goes from awake to deep sleep quickly then sleeps deeply all night and hence rarely wakes.
On the rare occasion she does wake in the night, I'm guessing she's difficult to get back to sleep? It's because she's not tired enough to go quickly into a deep sleep, and isn't tolerating the light sleep phase.
Daytime naps are always going to be less deep sleeping than night sleep, which may explain her struggle with going to sleep and also staying asleep in naps.
Sounds like your answer will come in understanding her fine balance between being very tired but not over tired.
The more tired she is, the quicker she'll go into a deeper sleep. But as soon as you cross the line into over tired, then she'll get restless and it all gets messy and stressful.
This is a bit alien for me, I'm talking hypothetical here. I am used to working from the angle of getting baby to sleep at the "just tired" point, with a gentle lead into light sleep and then leaving baby to go from light to deep sleep themself.
In your DDs case, you want to skip as much of the light sleeping phase as possible. She wants to be so tired when she goes in the cot that she drops into a deep sleep quickly. (Like my husband, snoring as soon as head hits pillow).
But... If you keep her awake too long, you will end up with a screaming battle. So it's a fine line.
Given all that waffle, how about keeping her active and awake longer than you have been during the day.
You know her better than me. It's about finding the point where she's past tired and into the very tired phase, but before over tired. That's going to be a difficult thing to balance but once you find a routine that works, stick with it.
If she has a good night, try for a single lunchtime nap. If she hasnt slept well over night, try for 2 naps with, say, 3h awake time. These are all just me guessing, you will know better the timings that will work. If 3h awake is making her over tired and fight the nap too much, try 2.5h. It will all be trial and error.