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Where mail addressed to a domain is delivered: every exchange in preference order, with the addresses and reverse names behind it — and the configuration mistakes DNS will never report on its own.
An MX record names the host that accepts mail for your domain, and a preference number that orders your hosts. A sender resolves them, sorts by preference, resolves the addresses behind the first, and connects. Everything else about receiving mail follows from those two lookups.
MX decides where mail to you goes. What happens to mail from you is decided by SPF, DKIM and DMARC — which is why a domain with perfect MX records can still have every message it sends land in spam. The full report covers both.
The preference. Senders try the lowest first and fall back to the next when it does not answer. Equal numbers mean the load is shared between those hosts, not that one is a backup.
It must not. RFC 2181 requires an MX to name a host with an address record. Most senders follow the alias anyway, which is why it survives for years; the ones that do not defer every message, so the symptom is mail from one particular correspondent never arriving.
Usually yes: a sender with no MX to use falls back to the domain's own A record. It works, it is invisible in every mail client, and it breaks the moment you move the website to a host that does not run a mail server.
A single record reading 0 . It declares that the domain accepts no mail at all, so senders fail immediately rather than retrying for days. It is the right thing to publish on a domain used only for sending, or for nothing.
Not directly — MX is about receiving. It matters for sending because a domain that cannot receive mail cannot receive bounces or replies, and some receivers treat a sending domain with no working MX as a signal in itself.