Maybe they're hoping they can keep you out until after the UK leaves the EU.  The decision is, as you have laid out, completely illegal and noncompliant with the UK's own guidance.

To answer your questions:

> 1. Should I include the refusal letter of the first application indicating the mistake (if indeed it is a mistake) that the fact that we have two kids together has been overlooked?

Yes.  I wouldn't discuss whether it is a mistake.  Just point out that you have two kids, and that their guidance says that this means your marriage is not one of convenience.

> 2. Should I include family photos stretching back the last 17 years or so?

No.  Prove your kids' existence with their birth certificates.  If you give superfluous evidence like photographs, you won't help your application, and you could hurt it if they find something that appears "inconsistent."

> 3. Will screenshots of the conversations help?

No, see answer for #2.  Keep it simple.

> 4. Should I include a family certificate?

What is a family certificate?  If you mean your marriage certificate and your childrens' birth certificates, then yes.

> 5. What else can I include to strengthen my case?

You should include nothing else.  Your case is simple (not to mention ironclad); keep the application simple.  Prove only the facts you need to prove:

 - You are an EU or EEA or Swiss national
 - You're married to your wife
 - You have two children with your wife

In conclusion, another reason not to add additional evidence is that it makes it more likely that the ECO will miss the crucial evidence: the legal documents.  The ECO should thank you, too, for not including extra evidence, because you will have reduced the ECO's workload.

You may want to consider appealing the decision concurrently with your new application, in case the new application is also denied.  There are stories floating around on the internet where multiple EEA FP applications were denied.

Before you appeal formally, however, or even reapply, you might want to make an informal complaint to the British consulate (specifically to the Entry Clearance Manager [ECM]) calling attention to the fact that the the ECO has not met the burden of proof to establish a suspicion that your marriage is one of convenience, and that in light of your two children, the decision doesn't comply with the guidance.

(The presumption that you should have presented evidence substantiating a genuine marriage when you've been married for nearly 2 decades is preposterous, even in the absence of children.)

For an example of this approach, have a look at http://www.immigrationboards.com/eea-route-applications/refusal-of-eea-family-permit-reapply-or-appeal-t106017.html.

You may also want to read [these](http://expatriates.stackexchange.com/a/7332/2440) [answers](http://expatriates.stackexchange.com/a/7626/2440) of mine and follow the links in them.