Can I write a programm on Linux "Text Editor" ?

Answer 1

vim and geany are good editors 👌

Answer 2

Yes you can!

Answer 3

yes you can write a c++ program on the text editor.

then you can compile it using the gcc compiler.

keep in mind:
you have to download and install gcc.

Answer 4

Yes,
editors like gedit and vim support syntax hightlighting for a lot of languages.

In addition, you can configure language-specific options, such as indentation width, replacement of tabs by spaces, etc.

If you like to use vim, consider to have a look at http://vimawesome.com for some plugins that make your life easier by integration of version control, etc.