How can I map some keys to those icons I marked in attachment? I would like to have them mapped to “1” for vertex, “2” for edge and “3” for face, but I don’t know how those icons are called.
And also I would like to map that three icons to the left of “local”.
Thank you, that’s it 
i dont recomend you to rebind those, its much better to just get used to it.
the shortcut for swiching between the three modes is CTRL Tab.
if you REALLY want to rebind them, type CTRL + Tab and then RIGHTCLICK the three options, and on the top, click Set Shortcut.
again not recomended, you should get used to using Ctr+Tab to swich between the modes, but there you go.
also, you have to be in edit mode while doing this. and in order to save it, go to file, user preferance, and on the bottom hit Save User Preferance after rebinding.
Why is that not recommended? I don’t have anything binded to those keys?
default is swiching layers with the numbers, witch is very handy later when you have a huge scene. being able to swich between the layers fast. also there are a few keys that are not bound to anything, and has no shortcut, those could be bound to numbers. i always have layer 1 open so i rebound smooth vertex to 1 for instance.
also you dont save any time rebinding them, and you should try to adapt to the softwares default unless you use multible softwares all the time.
me for instance use C4D and blender 50% 50% ish, but i still try to stick to each softwares defaults.