I’ve seen this episode in syndication several times, where Robin has 4 dogs given to her by ex-boyfriends. Is this the only episode (as I suspect) that mentions her dogs?
No. In the pilot, before we know that the dogs were given to her by other men, one of the first things that Ted knows about Robin that makes him fall in love with her is that she loves dogs. He goes to her apartment, and the dogs are in the scene - Ted is scared when he actually meets them.
I’m not sure if they appear in other early season eps.
I ask because she seems to have a lot of time to hang out in a bar for a career woman with 4 dogs.
Be fair, we never see any of the characters at the bar for more than a few minutes at a time. 
She hosts a four AM news show (or she did, not sure what her current gig is). That gives her loads of time during the day.
Robin eventually sent her dogs to live with her lesbian aunt on a farm upstate.
In the episode where Barney bets her to say stupid things on the air for money she gets called into her bosses office. She thinks she’s in trouble, but it turns out even he doesn’t watch her show…he just wanted dog advice because she has them.
Also, besides working at 4am, and hanging out at the bar, it seems like they are really only at the bar for a combined total of an hour or two a day. It’s not that they all get there at noon and don’t leave until 2am.
In a lot of the scenes they seems to be just getting there or just about to leave. Also, it’s fair to assume that in any scene that Robin isn’t in, she may be at home.
In the first few seasons, she is an “on the scene” reporter and then an anchor for “channel one” news (the joke being that there is never a channel one.) She doesn’t become a morning show host until season 4, after she moves back to NY from her very brief stint in Japan. Her current gig is actually just a researcher for a national news network, no more being an “on the air” personality.
She actually gives up her dogs in the second season because both her and Ted agree to get rid of all the things their exes have given them…though Ted re-negs on the deal and keeps things because he doesn’t believe Robin would actually give up her dogs, but she does. Unlike when your parents said your dog went to go “live on a farm,” it was actually true in Robin’s dogs’ case.
They were pretty consistent with the dogs.
In one episode I remember her leaving and telling the dogs to stay out of the liquor cabinet.
In another, Ted is going to make a grand gesture and he hire an orchestra to come and play with blue instruments - he borrows the spare key from Lilly. Robin comes home and finds she has forgotten her keys and calls Lilly. Lilly says: Just knock. Robin says: My dogs are well-trained, but their not *that *well-trained.
And no-one watches it.
NY1 is a genuine local cable news station in NYC.
Come to think of it though, I don’t think the dogs were actually in that episode. Of course, I suppose we can assume Ted could have locked them in the bedroom for the evening.
I have seen a bunch of episodes, but this is the only one I’ve seen where her dogs were mentioned.
Then I think you have your answer. Also, to go back to your second post in this thread, don’t forget, dogs don’t need 24 hour care. I have 2 dogs that can easily (if need be) be left alone for 8 hours at a shot. But I typically just have to make sure that when I’m not home for long stretches I stop home every 4 or 5 hours to let them out for a little while. I think it’s fair to assume that even if she’s spending the day at Ted and Marshall’s she may still stop at her apartment once or twice to take them for a walk and feed them. Also, like I said earlier, we see the group together a lot but since she works early in the morning, I think it’s fair to assume that if she’s not in a particular scene, she may be at home.
Also, another thing to keep in mind. the gang meets Robin in the pilot and she gets rid of the dogs midway through the second season. So she only had to keep it up for a little over a year.
they’re not, dammit, they are not.
She’s Canadian! English might be her second language!