I believe we should stop telling folks how to manage their feelings and just answer their questions about their pets.
Tolerance is key
The site is built upon the Be Nice policy.
Unless it is supportive, there is no reason to be telling people to calm down. Pets are an emotional area for pet owners. People can be as attached to their pets as they are to relatives, children. When someone is distressed we need to be mindful of how they are feeling and adjust our tone accordingly to soothe that person.
Not everybody possesses this skill. So the key would be, if you are not so good at being empathetic, or just having a bad day and feel intolerant, don't comment. There's plenty of posts where the OP is not emotional.
Where do we draw the line with emotional users?
Now there's blurred lines of giving factual information and emotional support on this site. Personally (and this is purely my opinion - it is not a community consensus), I believe in offering people some emotional support.
My limits are:
1. can I offer this person some support within the comments? yes, I offer support.
2. if it would require to go to a chat room - redirect the person to a place to get more intensive emotional support.
This line of emotional support needs to be made with each individual participant of the site. No one should feel forced to comfort someone, as that's not the nature of these sites and feeling this pressure could alienate many users.
The simple thing, it's an option for users to do this. If you feel unable to help and know there's users around with a history of helping with difficult situations, ping them in chat or under the post. You're not alone, we're a community.
Supporting the impossible situation
Gender issues aside. Many people here male and female get distressed about their pets. The gender issue is an issue for some Stack Exchange sites, I haven't noticed it being an issue here, as we have a nice balance (feel free to correct me if I'm wrong - I've been away for a while). Some people may be prone to react more to being told to calm down, either way it's not helpful and doesn't encourage people to want to use the site.
A simple example of someone in distress is shown in the comments to this question and answer. Now I am probably more open in some ways than most people would want to be. But in this post, I've had to give bad news - an animal will die without treatment and the OP has no way to get the animal treated. How do we deal with such a tragic situation?
What is the signs that a lonely kitty is abandoned by the mother
We're a global community
This brings me to my next point. We're a global community and must be mindful that not all cultures have the same attitudes towards animal welfare. They have different styles of relating and writing. For some people English is a second language and they may be using a translator to write. So with all these variables, how a post may be written and how a person is actually feeling, may not be consistently interpreted over the internet without clarification. For example: Many people will use excessive punctuation, which can be a cultural thing.
Also not everyone has the financial means to get veterinary care. It's most important not to berate people for this, but to be supportive, without denying the facts. Now this is the most difficult task of this site (for me) and not one that everyone will be able to excel at, but practice makes purrfect.
Don't forget, other users are a ping away, if you're in doubt and want a second opinion of how to handle a situation.
TLDR Don't be telling people how to manage their emotions, just be supportive.