First and foremost HR IS NOT YOUR FRIEND
The unfortunate truth is that you are, in fact, violating policy by being late and being absent and there is no legal protection against that.
To assume that an employer will be compassionate is to set yourself up for failure.
Your employer does not care that you are going through hell and they are not legally obligated to make life easier for you. The fact that they extend a courtesy to others does not mean that you are being discriminated against, it only means that you are not being extended a courtesy. Part of that is likely the fact that you are a new employee. Someone who has been there for 20 years has a history with people and will get leeway that you won't. That isn't discrimination, that is simply life.
Now, to address your core concern. Since you have no family, go to friends, go to a church, temple, charity, ANYTHING you can find to help you with childcare. You need to find a way to make it so that someone other than you can handle emergencies during work hours.
Get the process going with church, temple, mosque, friends, groups that can help you and then talk to HR about how you are working to make your home life more stable and ask for leniency on that point.
Ultimately, it is up to you to make sure that family issues do not affect your job. It is an unkind fact, but a fact nonetheless. You owe it to your children to take any and every step you can to preserve your employment. Don't hope that the employer will care, because they don't.