Show current file size in vim editor

This will work if you install both the airline and tagbar plugins. These two plugins integrate automatically and you'll get the current function displayed in the status bar. If you want to have the full object hierarchy (e.g. class + method), you'll have to configure that in your .vimrc:

let g:airline#extensions#tagbar#flags = 'f'  " show full tag hierarchy

You might have to setup the tags file for tagbar to identify the current code position, if you are not using ctags, yet. If you don't know about that yet, here's a short intro to ctags.

As Daniel Andersson says, a syntax file will get you a long way towards what you want. It's still not the same as a WYSIWYG editor, but headings, bold or italic text and code will look different if you have a proper syntax file set up. Note that there are a few competing Markdown plugins. In addition to the one Daniel mentioned, you might try this one or this one.

But in addition, you could try a plugin for Vim like WriteRoom for Vim. The idea is to make Vim visually distraction free. If you add such a plugin, and make (g)Vim as large as possible, you may get even closer to what you want.