I need the reason why we should not install SQL server on the domain controller
The points you have listed are shown in BOL here, and even though it quotes:


  SQL Server Setup will not block installation on a computer that is a domain controller


It states on this KB article that the setup will fail, although the applies to only list up to 2012 version.

One main reason you are going to have against doing this is Microsoft Support. As it states in documentation you should not install it, they are not likely going to provide support. Especially if you experience any security or performance issues. Unless you fork over a good bit of money they are probably not going to do much for you. 

I have never seen a company put these two roles together out of choice. I hate to say it but if they need to run an application in SQL Server and it is going to be a mission critical application there is no reason for them not to invest in running that application. Hardware is cheap these days and it does not take much for AD to run if your organization is that small. You could even go and get 2 or 3 year old hardware for that matter.

Now with that said, Window Small Business server used to be the one stop shop for a situation like you describe. It is an AD, Exchange, and web server all on in one; and if I recall also had an installation of SQL Server but I can't remember. I setup one of these servers for a department one time.

However with Window Server 2012 they changed this to Window Server Essential Edition, and I am not sure what roles you get with this Edition. I would suggest though you simply spend the money to get Standard Edition. A feature you get with Standard is licensing to run two guest OS virtual machines via Hyper-V. If you size the hardware appropriately it will allow you to run your SQL Server instance in one VM and then your application in the second VM. Although your performance is all on one server it provides a boundary separation and would be a fully supported setup by Microsoft. Just my two cents.