How should I handle this/is this normal?

It's a small linked-in world and you never know where people will land. Four days is not very long. If it were me and they pushed back I'd skip a couple of those days off. You can use whomever you want as references, but that doesn't mean they won't be able to talk to others, especially the boss.  I'd advise not burning any bridges.

Umm, you do realize that one doesn't follow the other.  Nothing to look at, and over 35 are completely unrelated.  I know some God-awful ugly 20 somethings, and some damn hot 30 and 40 somethings.

As for the OP, yes you screwed up.  Even before seeing what you wrote later about your departure being handled professionally, you really need to follow the conventions of your industry.  In most, that means 2 weeks notice (and sometimes more).  As an educator, if I gave a  months notice but left mid-year I'd likely never find another job in the field since the norm/expectation is to finish out the school year.  When I was in car sales, I gave 2 weeks notice to the general manager at my first job only to be yelled at for about 20min and then kicked off the premises.  A sales manager who I was friends with took me to the side and told me that if I was to stay in car sales I needed to know that you give your notice as you leave your last shift (I later learned that he did just that at the end of that day).  I was told that they are afraid you may try to take your customer lists with you and that you would have a bad attitude in your 2 weeks and lose them sales.  So, it can be industry specific, but in most professional positions the norm is either 2 weeks and can be 4 weeks (my current employer requires at least 4 weeks if you want a good reference).  Not following that can lead to a bad reference.  As someone stated, you may not plan to use them as one of your 3 references, but potential employers will likely contact the employer anyway.  One of the usual questions is whether you would be re-hired or not, and some employers will use whether you gave sufficient notice or not as part of their answer to that question.  Whether or not you left with sufficient notice can also sometimes be an interview question.