For the average filmed episode of a TV show, how long is the shooting day? How many minutes of an episode are filmed on an average day? Has this changed over the last 20 years or so? And are episodes filmed every week of a show’s shooting run or do they have breaks?
I worked at Nickelodeon Studios in the early and mid-90s, when the network was still producing live shows. I worked on multiple seasons of Goodburger, Clarissa Explains It All, Super Sloppy Double Dare, Legends, and at least 6 more shows that I can’t remember or be bothered to look up.
I worked on several episodes of Seaquest DSV as well as a short-lived series called Treasure Hunter or Fortune Hunter or something like that.
I’ve also worked on Jeopardy and Wheel Of Fortune whenever they come into Las Vegas.
So I have some experience with shooting days on TV shows.
The average shooting day is 10-12 hours. The cast is not usually on set (or even on property) all that time. An average day will result in anywhere from 8-24 minutes of usable footage. In my experience, this has not changed much in the last 20+ years.
Your last question is tricky. Sitcoms and the like will shoot part or all of an episode in a day, and complete shooting a season in something like 2-3 months. Game shows will shoot a weeks worth of shows in a single day, so an entire season of a game show is shot in just a few weeks, although rarely do any of them simply shoot 360 episodes and then relax for the rest of the year. Rather, they shoot a month’s worth of shows in a week, then promote the show and relax for 2-3 weeks, then do it all over again.
A 30 minute sitcom will take 5 days to shoot, averaging 4-6 pages a day. It takes 8 months to shoot a season.
I was friends with a guy who worked on the production crews for Star Trek: The Next Generation and Star Trek: Voyager. From what he told me, their production schedule (to put out ~26 1-hour episodes for a season) was about 9 months long, though they weren’t shooting for that entire time. They’d start production for a season in August, with the first episode finally premiering in late September. They’d start out with a reasonable cushion of time between production and air dates, which would slowly get whittled away through the season, with shooting finally wrapping up sometime in early to mid May (and the final episode airing in June). The crew would then have several months off, until production ramped up for the next season. As I understand it, they’d rarely have time off (other than some time on weekends and holidays) during the production season.
By the time they would get to a season’s last episode, there would often be edits and SFX tweaks being done in the final days before airing.
As far as shooting goes, they would be shooting on most weekdays throughout most of that production schedule, though I don’t know what a typical shoot day looked like. I do know that Genevieve Bujold was originally cast as Captain Janeway on Voyager, but Bujold (who was primarily a film actress, rather than TV) was not prepared for what the shooting schedule for a TV show would be like, and she quit after just a few days of shooting.
The show my daughter was on was 30 minutes (so actually 25 or so) and took five days to shoot. Since it had kids they were limited in how long each worked during a day, but the crew did 10 - 12 hours easily. We usually had lunch and dinner on the set. One shoot which was exterior night went to midnight, but that was not the rule. It was all done on location also, which took more time no doubt.
I’ve heard from many Star Trek alums of various generations that the shows were grueling - often months of 10-12 hour days for the crew and headline cast, especially as the schedule slipped, as kenobi_65 notes. IIRC, it’s a running theme in Shatner’s semi-documentary Captains.
Sort of a related question: Michelle Dockery says she’s tired of the 7 month shooting schedule for a series (season to us Yanks) of Downton Abbey and will only do one more season. (Yet another cast member bailing.) That’s for like 8 episodes and a Christmas special.
Why?
A US drama will shoot an episode in a week, two tops. Filming for a month or two, taking a break, then resuming. And they can do 22-23 episodes in 7-8 months.
What is the deal with Downton?
The bulk of a The Big Bang Theory episode is shot in long afternoon and evening. Some stuff may be filmed ahead and shown that day for audience reaction. Re-shoots at a later point sometimes happen.
For a typical US sitcom there’s a schedule something like: Monday- table read. Tuesday- rehearsals, work on script repair. Wednesday- full set rehearsals, shot blocking. Thursday- shooting. Friday- re-shoots, leftover scenes.
And really long days are the norm when shooting.
At a guess, really limited shooting hours at Highclere. I know the Earl’s family departs when the film crews arrive, and I’d gather that for all the benefits of being so highly featured, being chased out of your house is a PITA.
Other locations maybe just as problematic. Then there’s that weather thing. I think shows like Downton shoot on every sunny day there is. 
That may be true for sitcoms that are “filmed in front of a live audience”. Your “typical” sitcom now shoots partly on stage and partly on on location, with no audience or laugh track (Modern Family, Parks and Rec, etc.). These take generally 5 days to shoot; the crew will work more than 12 hours many days (the drivers have the longer days, usually 14-16).
A further thought: It may not be that they do full-cast shooting for seven months as much as that the cast is required to be available on short notice for some amount of that time that isn’t set-call days. That is, she might shoot her role in a few days for each episode, but be required to report to the set or studios with little notice for the entire 6- to 7-month period, severely limiting both her personal and professional options.