What I didn't understand about these pings is wouldn't their wavelength be massively lengthened by the time dilation? How did they even receive Miller's signal if it was redshifted ~60000 times?

That is an excellent point, I didn't even consider that. It's possible they were listening on the correct frequency to receive it, since they knew the redshift was going to happen, but that just leaves more questions as to why they didn't then realize that it definitely couldn't repeat.

But they could. That is exactly why they did know about nice atmosphere and water on the planet, but didn't know about the waves since the first wave that killed first astronaut happened during the landing. They also left one of the team member to do gravitation research and planned to spend as little time as possible on the planet.

Yes, they knew about the relativistic effects. And they could obviously send some data. But somehow they can't figure out that they've been getting a repeat of the same ping for the last 10 years? And even if they couldn't figure that out, how could the person on the planet possibly send a ping every year, if she only spent an hour and half on it according to her frame of reference?

I don't remember how many pings they received in the movie. I've understood that they had received just one ping from that planet.

They expected a ping every year, if it ever stopped pinging they would write off that planet. It was up to the scientists to keep the ping going, if they considered the planet still a suitable candidate for humanity. The signal they received from the time dilation planet was supposedly "looped around the black hole" and repeated yearly.

If I remember correctly, one of the characters hand-waved it as (paraphrasing) "multiple echoes of the same signal". Meaning there were no meaningful timestamps

Ping issues were especially bugging me. Take those pings from the relativistic planet. First it is believed that the person has been sending data for many years, yet only after landing do they recall relativistic effects, and that in fact that person has spend only a couple of hours at the surface.

The biggest issue I see is the fact that gravitational time dilation is considerable on the surface of the Miller's planet, but not so much in the relatively close proximity of the planet, where the mothership is. I just don't get that part.

The explanation Kip Thorne offers in his book is that the lander ends up slingshotting around two things, like black holes and neutron stars, that happen to be in the right places to drop the lander into the gravity well and then accelerate it back into a stable orbit close to the planet. Of course, that kind of maneuver twice, on both the descent into Gargantua's gravity well and the return trip, is totally implausible... the idea that there are so many high gravitation objects orbiting Gargantua near the planet's orbit that you can pull it off on no particular schedule... but it might not be physically impossible.

Wouldn't that be yet another reason to avoid that planet like the plague? Even if they set up a colony there, at any time the planet could run into an orbiting black hole or neutron star, or get slingshotted by one of them into an unfavorable orbit or into Gargantua itself.

I figured the mothership was still orbiting the black hole, in a much wider orbit. They could rendezvous by matching orbits using pretty standard manoeuvres, it would just take more time than if it was moving in a similar orbit as the planet itself.