Partial, with related page info, some of which only came to light recently:
NASA had a live broadcast on November 21st that was really interesting:



Really nice to see the Moon growing in the view in real time.

https://twitter.com/Explorer_Flight/status/1594647403243896833
Discovered yesterday that DSN 34meter dishes can handle greater bandwidth than previously analyzed so this will allow more live video streaming and imagery downlink. http://NASA.gov will start carrying the feed at all times (sorry there will still be some gaps).


- Also, with regards to footage of Earthrise, live or otherwise, there was an update of this less than 58 years ago. In the period 2007 to 2009, the Japanese space agency had a multi-spacecraft probe sent to orbit the Moon. On board was a couple of HDTV cameras which sent back footage. Some of the footage included some very nice Earthrises.
- Added info at end of answer.
https://www.nasa.gov/feature/nasa-s-artemis-i-cameras-to-offer-new-views-of-orion-earth-moon
Nov 16, 2022
NASA’s Artemis I Cameras to Offer New Views of Orion, Earth, Moon
There are ... 16 ..cameras.. on Orion – to document essential mission events including liftoff, ascent, solar array deployment, external rocket inspections, landing and recovery, and capture images of Earth and the Moon.
Four cameras attached to the spacecraft’s solar array wings on the service module will help engineers assess the overall health of the outside of Orion and can capture a selfie view of the spacecraft with the Earth or Moon in the background.
“Each of Orion’s four solar array wings has a commercial off-the-shelf camera mounted at the tip that has been highly modified for use in space, providing a view of the spacecraft exterior,” said David Melendrez, imagery integration lead for the Orion Program at NASA’s Johnson Space Center in Houston.
Due to bandwidth limitations on the spacecraft that prioritize transmitting critical data to the ground, livestream video quality will be lower than the onboard recordings. As a result, some of the highest quality views may not be received until well after they are recorded and can be downlinked.
The field of view of each camera has been optimized to look at the spacecraft, not deep space










https://www.foxweather.com/earth-space/artemis-1-nasa-orion-spacecraft-live-views-of-the-moon-lunar-orbit
As Orion began its closest approach to the moon yet, NASA shared live views of the moon and Earth as seen from the spacecraft. Orion will complete an outbound flyby burn Monday to enter lunar orbit and make its closest approach to the lunar surface at just 81 miles away.

As Orion began its closest approach to the moon, NASA once again shared live views of the moon and Earth as seen from the spacecraft. Early Monday, Orion was more than 230,000 miles from home, quickly closing the gap to the moon. The images were provided by cameras on Orion's solar arrays or the spacecraft's "selfie sticks."

https://www.foxweather.com/earth-space/the-eyes-nasa-orion-spacecraft
Redwire Space camera system will also help provide live views of Orion's journey to the moon and back.
Redwire's camera system will also help provide live views of the journey to the moon and back. There is also the opportunity for stop-motion photography.
"We'll be able to see the entire journey from the Earth to the moon sped up," Bailey said. "But the real-time parts of it … having video, especially if we can get live video of lunar orbit insertion …. and just orbiting the moon and see and seeing the spaceship with the moon in the background, there's going to be some tremendous photo opportunities here."
Edit to add HDTV info from Kaguya:
SELENE - Selenological and Engineering Explorer, also known by its nickname Kaguya, was a Japanese lunar orbiter spacecraft in operation from 2007 to 2009, in a polar orbit. On board was both a telephoto and a wide-angle HDTV camera, 50.1 degrees wide-angle and 15.5 degrees telephoto. One minute video scene is compressed, stored, and transmitted to the earth through 20 minutes.
kaguya HDTV footage looked like this:
Wide angle (15.5 deg): 
Telephoto (50.1 deg) : 
Two Earthrises taken in 2007 and 2008:
KAGUYA (SELENE) taking "Earth-rise" by HDTV on Nov. 7, 2007. Aspect ratio 16:9 and HD
KAGUYA (SELENE) taking "Full Earth-rise" by HDTV on Apr. 5, 2008. Aspect ratio 16:9 and HD quality
Given the above was achieved around 15 years ago and the DSN has been upgraded since then, live (standard 1080) HDTV broadcasts are achievable, at least at this distance.
Possibly was not live, but here is an Earthrise from Orion:
Nov. 21, 2022 – Earth rises from behind the Moon in this video captured by a camera on one of Orion’s solar array wings. The video was taken at 8:05 a.m. EST on flight day six of the 25.5 day Artemis I mission, shortly after the outbound powered flyby and six minutes after the spacecraft regained connection with NASA’s Deep Space Network