No, why would it? It will run code in the context of the current user which is absolutely enough to start a new process that will run in the background, download more code from a attacker server and allow remote access. The attacker will only have as much permissions as the user executing the code but that is enough to steal their files, run a keyloggers, steal their sessions for other websites etc.
They can try to escalate to the admin user, but when targeting private victims, all the data that is worth stealing is available to the user and does not require admin privs.
.LAN is not an official top level domain. So I assume this is either your home network or work network? In any case your problem has nothing to do with the .LAN doman.
Maybe you have “https everywhere” activated. If so, Firefox will always default to https unless you specify http in the URL. Again, unrelated to .LAN.
For the certificate: what do you mean “when available”? A self signed cert is a self signed cert. There is no “available” or not. You can import the certificate into the Firefox trust store so Firefox will trust that one specific cert but any other self signed cert will cause an error. That is expected and save behaviour (and unrelated to .LAN).