I will use arduino to make tasks in my life automatic maybe to let rotate every morning a motor or to open the door.
That's the kind of thing it's designed for.
So i will ask can i run arduino as solo so no pc needed,
Yes, you only need a PC to program it, once programmed it will run without the PC.
Can I connect it to the internet
Yes, there are Ethernet and WiFi shields available for connecting it to a network.
and can I setup PHP and Apache on it to control it over it; has it any protocols like SSH like an normal server ?
No. The Arduino isn't powerful enough to run software like that. You would normally connect it via the network / internet to a server to process data. It would just receive simple instructions or send data to the server for processing.
as example: I will send over any protocol a signal to my Arduino with a certification and then the Arduino is running a function...
I assume by "a certification" you want to run SSL on the Arduino. That's not going to happen since it's way too underpowered for that. As I mentioned earlier, the Arduino would connect to a server on the internet to receive its instructions, or you could interface to it directly within your local network (say from a smart phone) through a simple web server interface (very simplistic due to lack of memory - a few buttons and text, that sort of thing).