It is a special use of the verb "mít" which works either as:
Nemáme nikoho, kdo by nás vedl. - here we are the subject of the sentence and we do not have anyone, so "nikoho" for the object.
Nemá nás kdo vést. - here the unknown non-existing person is the subject, so (ni)kdo for the subject
So is být used to talk about what there is or isn't only with things and not people? Can you use (ne)mít to talk about whether or not there is/are a thing?
Or is the usage of mít based on what comes after, with the idea being that one needs to "have" somebody to lead them?
No, být is used with both. And the mít construction as well.
Nemáme nic, co by...
Není nic, co by...
Nemáme nikoho, kdo by...
Není nikdo, kdo by...
With things, the "Nemáme nic" is really about what we have available or what we own. "Není nic" is about existence of such a thing.
I heard and put "Nemá nás kdo ves", which was accepted, (although it should not have been) and then reported it as "audio was wrong" because that is what it sounded like to me, and I played it again several times at both normal speed and slowly, to check what I heard. Is it normal to drop the consonant at the end of a sentence in Czech? I know in English there are lots of dropped letters but Czech so far seems generally phonetic.
I hear "st" at the ending. It is not strong sounded, but still quite distinguishable for me. I do not think "t" sound is dropped here, just the intensity of pronounciation fades approaching the end of the phrase. Try to pronounce fluently yourself: "kdo ves" and "kdo vest". Maybe you will catch the difference.
It is not dropped I can hear it clearly. It may be a bit quiet but it is there. A different user had a similar issue with K at the and of a word. Both T and K are voiceless plosives.
Note that someone also reported that the voice sounds differently on different ios devices. I am using a web browser.
I'm using Firefox browser on a Windows desktop, with some reasonable Bose loudspeakers. Music sounds good but I have to say Duolingo comes over a bit fuzzy at the best of times. My wife is doing the same course on an iPad, the sound there is also a bit fuzzy, she has had a number of issues but does not discuss or report because the device throws her out of the lesson if she does, and she has to start it again. Can I ask, what is a Voiceless Plosive?
(ex)plosives are sounds generated by letting the air out by releasing some pressure. Voiceless means there is no sound generated by the vocal chords. http://www.fon.hum.uva.nl/rob/Courses/InformationInSpeech/CDROM/Literature/LOTwinterschool2006/speech.bme.ogi.edu/tutordemos/SpectrogramReading/cse551html/cse551/node36.html