and wanting a "basic view of the stage" + "encoding to combine audio" are not "entry level" items.
At that distance, the stage will be far away; "zoom" or getting a closer view (likely to fill the monitor) is not something inexpensive devices can do. There may be a security camera with an add-on lens possibility...
We don't know which TV or monitor is being used.
AppleTV, Chromecast and other similar IP media streaming devices connect to the monitor with HDMI and an IP LAN with Ethernet (AppleTV) or wifi (all of them). The video source needs to be on that LAN and streaming-device compatible. That means either a DVR or NVR is connected to a camera and the DVR or NVR is somehow connecting to the streaming device (wifi or Ethernet cable to wireless access point) compatible... or just the camera is connected to the LAN and is somehow streaming device compatible. I am not aware of any DVR, NVR or camera that meets this requirement. That does not mean they do not exist, merely that I don't know about them. I am 99% sure my AppleTV (v3 & v4) won't deal with this configuration - but others may - I might be totally wrong.
If you went the security DVR/NVR route, most have either HDMI and/or VGA monitor connectivity. So... no IP streaming device needed (just cabling long enough to connect all this together - which reminds me, we have no idea what the distances are from the camera to the monitor. Most DVRs and NVRs have a separate mic input for audio, so it takes care of your other issue. We also don't know if cabling is even possible - there are various "wireless" video/audio options - and none are inexpensive (especially if we have non-line of site issues which require more power and are lots more expensive than cabled).
Check the DVR rear panel image. HDMI and VGA out for video. Audio input for a mic and output for a feed to the monitor's RCA audio input. This particular system uses standard definition composite cameras. Just because it has 4 camera inputs does not mean you need to use all of them.
In the case of these specific cameras, assuming the TV/monitor has a composite video input and audio input, you can just plug the camera into the monitor - for your application, no DVR needed... and add a power (plug into the wall) for the camera. All cabled... At worst, you would need to add a BNC-RCA adapter.
Check the NVR rear panel image. Again, HDMI and VGA out for video. In this case, the camera's have a built-in mic. I am not clear if the audio out is needed to connect the NVR to the monitor's RCA audio input or if HDMI out will carry audio with the active camera's video (this should be a quick call to the manufacturer's tech support - or email or read through the manual). Please note this is all 1080p HD. The distance the internal camera mic can be effective says 16 feet - while this does not meet the 80 foot requirement stated, note that this is 16 feet from an audio source. Put the camera next to a speaker - or add a speaker to the PA and placed near the camera. Again, use only 1 - not all - cameras. Note that this system uses CAT5e or CAT6 data network cabling between the camera and NVR. Both the camera and NVR are IP network elements and have their own individual unique IP addresses. Plugging the camera directly into a monitor with an ethernet port probably will not work the way you want. The camera configuration is problem 1 while problem 2 is the camera needs to get power using PoE (Power over Ethernet) which the TV likely will not provide.
These security camera systems are designed to be on 24x7x365, so getting the TV image per your requirement ("to be as simple as turning on the TV and seeing the what's the camera sees") is met.
In both cases, if you decided to add the DVR or NVR to a computer LAN that has internet connectivity, you *could* allow limited remote internet access to those with a password so they could watch from home or smartphone (assuming that was not a lot of people which could crash the system - it is not built for lots of separate, remote, viewers... this is a security camera system, not broadcast or streaming service).
The missing item in all the above is the "zoom". Typically, a "varifocal manual zoom" lens is more common in the analog versions using the composite BNC/RCA connections (like the first example). Under certain conditions, a magnification lens can be added to a security cam if it has the appropriate lens barrel mounting screw. Of course, being able to place the camera closer to the stage makes all this zoom discussion moot.
Another possibility is a camera that has controllable zoom. These are usually only the PTZ types (where "Z" = zoom). You won't need to use the the zoom control after it frames the stage to your liking. You may not use the other features, but this is not an "ala carte" feature - it comes with other features. There are pan/tilt only cameras. There are no zoom-only cameras other than those where the magnifying lens is screwed on.
For example, this PTZ cam has 10x optical zoom.
This one has 20x optical zoom
Before buying, contact the manufacturer for DVR/NVR compatibility (and confirm the audio will work as planned).
You can also contact various retailers that may be able to help with installation (Fry's Electronics, BestBuy, etc and get some "consulting work". It may not happen, but worth a shot. The alternative is professional installation. That can be exciting - as they normally deal with pro-grade studio gear for what you want to do (as opposed to making a security camera do the job of a studio camera - which is essentially what you want to do).
If wireless, bhphotovideo.com can usually source stuff, but way more info about the environment in which this set up is being installed must be understood.
Please understand, I am using Swann for examples, only. Lorex, Flir and others make similar gear. IP-network based cameras from manufacturer "x" will not work with IP-based NVRs from manufacturer "y". Heck, some IP cameras and NVRs from the same manufacturer won't work together... most of the DVR gear will work together even from different manufacturers.
Camcorders will be problematic. They are designed to conserve (battery) power and normally will auto-shutdown after a period of time when not recording video. There can be exceptions to this through setting options, but I think this route is asking for more trouble that it is worth.