There seem to be some misunderstandings. It might help to clarify what some things mean first, even if it becomes a bit long.
Lets start with some things a usual home router (in reality a combination of several independent things) can do for you. Very simplified:
- Being a modem
- "Translating" data from your computer to/from something that goes over the cable that leaves your house
- These types of signals are usually not the same, because of a) other applications on the same cable (phone, etc.etc.) which means your signal must conform to the kind of signals used by them, and/or b) different technical requirements (like having enough "power" to be transferable over several miles).
- Being a switch (if you can plug in multiple computers - and completely independent of any outside world internet connection)
- Being able to distinguish these computers (with help of MAC addresses).
- Making it possible that one computer sends something to another, leaving it to the switch to pick the right destination.
- MAC addresses are completely local, not suited for worldwide addressing.
- Being a router
- Being able to understand IP addresses (which are more than local), and deciding with them where something should be transferred
- Assignments of IP addresses can be like "this one IP belongs to this one MAC here locally", "this other MAC has this whole range of IPs" (maybe because the MAC is another router with several computers), or "all unknown IPs should go to there" (eg. to the modem and then the outside world, leaving it to your provider to find the destination).
- Offering NAT
- Instead of transferring data from the local computers to outside using the computer's original IP, the router sends the data to the outside as if it is from the router itself. Incoming responses go to the router (the outside doesn't know the computers behind it), and the router relays it to the right destination.
- Originally this was because there is a limited amount of IPs - all computers using their own (which must be unique worldwide) would be a problem. Now only routers need their worldwide IP.
- Between routers and local computers, a completely separate numbering scheme is used - still IPs, but ones that are never seen outside of your local network.
- Unintentional side effect: If someone "outside" wants to hack your computer, he/she must either hope that your computer contacts him first and then send a malicious response, or hack your router first to be able to send data directly to your computer. Because, as described, your computers don't even have global IPs - the only possibility to get data from the outside is if the router knows they want it. ... Of course in an ideal world your computer can't be hacked - but most can, and making them reachable directly for everyone in the world is indeed a risk.
- Something that filters/censors any data it gets.
- There are many kinds of criterias when to filter (of course not all FWs support everything), eg. "allow only if it is from/to these computers", "allow only emails but not browsing", "allow nothing where the word Frog is transmitted", etc.etc.
- In a non-NAT environment, a simple use case also would be to "allow only incoming things your local computers have requested before" (or in other words, only outgoing connections). As having the computer directly reachahble by the whole world is not good, this can be the same protection as NAT is. Especially useful as with IPv6 there is no shortage of IPs anymore, making NAT not needed.
So, to repeat a part: Allowing others to reach your computer even if you didn't go to their website etc. is not good, and a major security risk in todays internet. Having a LAN with NAT automatically protects from this. A firewall can protect too (most home users use their firewall only for this).
You explicitly told the technican that you don't want a router - meaning you have modem but no routing and no NAT, and you can only connect one device to the internet (even if there can be multiple plugged in - maybe you have a router-modem combination where the router is turned off per configuration).
As it is unlikely that a builtin FW will still work like this, you are reachable by the whole world - bad for security. If there is no reason to be reachable, don't do this (eg. web servers have to be, but there should be also continous effort to keep them secure).
What you can do if you want this protection: Ask them to enable the router again (or give you one if the device you have is a pure modem), or buy your own router to put it between modem and computer.
About your neighbor - if we understand "LAN" as the things inside your house, there is no way to sniff that without having access to the cables (I understand you don't have Wifi?). (This fact is independent of "router or not").
But it doesn't have to be inside of your house - as you have only one computer, all network usage will be to the outside world, and there are plenty possibilities. For example your provider. ... About your neighbor, if the provider is not completely bad he shouldn't be able to get access to your traffic data.
In any case, using HTTPS wherever possible goes a long way to avoid sniffing.
Finally, about wired internet plans - your connection to from the modem/router outside can be over cables (telephone cable, coax, ...) or not (cellular, satellite, ...). Having Wifi or not is completely independent of that - Wifi is about the part between you and the modem/router.