The real reason Karen didn't invite Will, Grace and Jack to her birthday parties was that they would stop taking her out to some crappy dinner, which is what she was really looking forward to about her birthday.
From the series finale: Sean Hayes and Megan Mullally performing a duet of Unforgettable. While it might be a bit of a Big-Lipped Alligator Moment in the episode, seeing as how Jack and Karen were paired off so many times in the show's run, it's a sweet moment.
In the revival, Karen tags along when Jack teaches drama to a bunch of middle schoolers, and winds up chatting with a girl named Tasha. The two get along in their own, snarky way, and it's eventually revealed that Tasha just had her first period and is feeling anxious about it. Karen, amazingly, turns out to be just the person to handle it, and comforts Tasha by telling her about her first period.
Karen: If I survived, so will you. Tasha: ...You're not really mean, are you? Karen:Mind your business. (she and Tasha smile)
Also in the revival, Jack meets his grandson, who's gay — and he finds out his son is sending the kid to a conversion therapy camp. Jack and Will promptly drop everything to rescue him.
Rosario's funeral is a combination of this and a powerful tearjerker. Especially Karen's speech at the end. Karen may be mean, but she truly loves and cares for her maid.