a) It really sounds a lot like a regular router (aka gateway). All features you've listed are the usual NAT (SNAT and DNAT) that routers are typically capable of.
For example, the 2nd item is regular DNAT (aka "port forwarding"), only with more specific match parameters – what you need is a router that allows matching on "external IP" in the DNAT rules, which many actually do. Similarly, the 3rd item is the same kind of SNAT (aka "masquerading") that many routers already do on the WAN interface, only in your case it's done exiting the LAN interface instead.
So the main thing to look for is whether the router's firmware is flexible enough, i.e. what you want is not the specific "home/office wireless" kind of router that assumes a fixed use case, but a more generic "enterprise router" that lets you build its behavior from scratch. (I don't know if that's actually the right term – it doesn't have to be enterprise-priced, of course – but really the problem is that it is just a generic router. Sometimes "enterprise firewall" might also fit.)
As a specific example, devices with RouterOS or OpenWRT should have the necessary capabilities. A computer with two Ethernet ports running an OS capable of routing and NAT (e.g. any Linux with nftables/iptables, or some BSD variant with pf) would also do the job fine.
b) That being said, everything could also be done with a "reverse proxy", the main difference is that proxies work at higher layers, e.g. instead of raw IP packets they deal with TCP connections or even individual HTTP requests, so you need to choose it based on what kind of communications you'll be performing.
But this actually means your 3rd feature is "built in" to all reverse proxies, as internally the proxy would establish a whole new connection from its own LAN address – instead, preserving the source address would be harder.
Reverse proxies also somewhat overlap with "load balancers" (many advertise themselves as both). They aren't necessarily dedicated devices or appliances, commonly they're just software running on a general computer such as Nginx, HAproxy, or relayd.
So if your requests are all HTTP-based, then an HTTP reverse proxy would work (it could even add HTTPS on the WAN side while still talking plaintext HTTP internally), but many proxies/balancers can do raw TCP as well. (I think generic UDP support is less common though, but not nonexistent.)
In both cases, I think "WAN and LAN ports" isn't really what you're looking for. I'd say quite the opposite – an enterprise router that doesn't have such ports pre-defined (and just has Ethernet1, Ethernet2, etc.) is more likely to be flexible enough for your purposes than a router which does.
