Hillary Clinton's age:
I would be the first person to say that age should be an issue when considering a president. It should be a part of the conversation. In that way, I don't think that bringing up age, in itself, for Clinton, is sexist.
It is the way her age is brought up that is sexist.
In 2008, when her age was far more typical for a presidential candidate than it may be now, Bill O'Reilly wondered if we really wanted to see Clinton age in office. If you wonder which angle, Bill O'Reilly is aiming for when he takes down Clinton, it's usually her gender he aims at. Do you think his insistence there "must" be a downside to a woman president is simply O'Reilly sexism, or is it aimed at Clinton? I would say both:
Rush Limbaugh brought it up, making the issue of her aging very gendered. Do you really want to see a woman grow old? Because men grow old so well, and my male presidential candidates should exist to please my eye. Lookin' at you, sweater vest, or as I mom called you, the "handsome one." *shudders*
As Ana Marie Cox noted on Up With Steve Kornacki, women are told they have an expiration date. We are constantly reminded of it. So what would an appropriate question on Hillary's age look like? See John McCain and Bob Dole.
Jill Abramson's firing:
As every writer and reporter covering this issue has said before, we don't know everything that happened.
I can't say with any certainty that it is sexist. As Jessica Valenti rightly noted on her blog, sexism does not announce it is there. It is slick, not oafish, most of the time, and that is why it's still around after all these years.
What sticks out to me about this is the way in which she received the boot. First we have a Politico piece that uses every gendered word in the book to describe Abramson's behavior. Then we have many of those same words brought out for a second rodeo when time came to justify her firing. That is fairly suspect. The only legitimate reason I am aware of for at least criticizing Abramson is making a decision without consulting the right people, i.e. Dean Baquet.
Even so, it doesn't appear to justify her firing. And then we have Abramson bringing in a lawyer to discuss pay equity.
You put all of those issues together, and then add to it the way that Abramson was "dumped":
The New Republic's Rebecca Traister explains in her piece, "Abramson's Firing Was Singularly Humiliating:
Abramson’s firing was among the most harsh and humiliating I’ve ever seen play out in the media's recent history. Within minutes of the editorial meeting at which the turnover was announced, Abramson’s name had been scrubbed from the masthead of the paper she’s run for the past two and a half years. A Times spokeswoman told Buzzfeed that Abramson would not be remaining with the paper in any professional capacity and would have no involvement in the transition of power. Sulzberger made no pretense that this was anything other than an unceremonious dump. When staffers reportedly expressed concern that Abramson’s firing would be a blow to women, he helpfully explained that that women in top management positions are just as likely to be fired as men in top management positions.
When you contrast that with the warm and fuzzy departure that Howell Raines received post-Jayson Blair, it is pretty hard not to come to the conclusion that sexism played a large role in this, even if sexism wasn't the only reason Abramson lost her job.