Not exactly, although to some extent this is so. There are lots of different nuances in when to use get and when to use have (and when you can or can't use either one), so learning how to use them idiomatically takes a bit of work. Let's see if I can unravel some of the tangle for you.
Get has the primary meaning of acquire, while have has the primary meaning of possess. (You get the feeling of "urgency" because you are picking up on the idea that acquiring something is more active than possessing it.) We often use them to mean pretty much the same thing, but there are some circumstances where they are a bit different.
Your second group of sentences are roughly the same, but with a slightly different shade of meaning. If you look at the primary meanings of the two verbs, you can see that "He got his days mixed up" implies that the mixing up happened at the point in time that the verb refers to, while "He had his days mixed up" implies that it happens at a time previous, and is the existing state at the point in time that the verb refers to.
This example should clarify the distinction:
I got a new car yesterday.
I had a new car yesterday.
The first one means that yesterday I acquired a new car. Perhaps I bought one, perhaps someone gave it to be, but I didn't have the car until after I got it. The second one means that yesterday I was in possession of a new car. This means that I got the car sometime before I had it. (Without any further context, the sentence implies that I no longer have the car, because if something happens in the past, by default it isn't happening now.)
These meanings run together sometimes. For example, I got lunch usually means the same thing as I had lunch. This is because having a meal is the same as eating a meal. (Again, with additional context, there are exceptions. For example, I could walk into the office with an armload of pizzas and tell everyone that I have lunch. In this case, I could also say I brought lunch and mean the same thing.) However, I got lunch can also mean that you paid the bill for everyone in a group, or prepared lunch for everyone, which I had lunch cannot mean. If there is no additional context, they mean the same thing. For example:
I got breakfast, so can you get lunch?
I got lunch for the kids.
The first one means that I paid the breakfast bill, so perhaps you will pay the lunch bill (or it can also mean that I made breakfast, so perhaps you can make lunch). The second one means that you made (or bought) lunch for the kids.
Now, your first group of sentences uses have in a slightly different sense. If you have Joe do something, you are requiring Joe to do it: My mother has me do the cooking on Sundays, for example. So, if you have someone admitted to the hospital, you are doing this on your authority, presumably because you are a doctor. If you get someone admitted to the hospital, you are persuading someone in authority to have them admitted.
Sometimes, this distinction is less clear, as in your third group. You got yourself certified because you are not the person with the authority to do the actual certification, and you had yourself certified because you complied with the rules by which a certification is conferred, so the person with the authority is compelled to obey the rules. So they have much closer to the same meaning. Here's another example of this:
I had him arrested because he broke into my garage.
I got him arrested because he broke into my garage.
The first sentence implies that you told the police to come and arrest him, while the second one implies that you asked them to do so. If someone broke into your garage and gets arrested, the effect is the same whether you had him arrested or got him arrested.
One more example:
I'm having the car fixed.
I'm getting the car fixed.
The first sentence means that you are taking your car somewhere and someone else is fixing your car. The second sentence could mean that, but it could also mean that you are fixing the car yourself.