I apologize for taking so long to reply and having to make you wait, the fault is entirely mine.
Before I get to your questions, let me just begin by saying that using Vimeo to host your videos, is the best option you can make for a number of reasons. When your site begins to receive traffic, streaming those videos will eat your bandwidth very quickly besides slowing down your site significantly especially with 18 other sites on your server.
You wrote something about slow streaming from locally hosted videos, and this is just the beginning. Everybody is slowly migrating to Youtube or Vimeo, so instead of doing that next year it would be much easier to start the new site off on the right foot. (Sorry for the lecture, but it had to be said).
On that note,
1) Please check on your server, if .htaccess file exists at the root of your website. There should be one, so please look inside for the lines below, and if not present, please add them (dont add in between open <IfModule and closing </IfModule). Please use a text editor to open the file.
AddType video/ogg ogv
AddType video/mp4 mp4 m4v
AddType video/webm webm
Save the .htaccess file.
2. From what I see the problem is that on homepage, there are at least 6 files ( itservices.mp4, xeo.mp4, 7tage.mp4, gans.mp4, dlan.mp4, seviceakademie.mp4) which are towards the bottom of the page, and are located where the image 'cover photo' is located on previous videos on the upper portion of the page. This causes the videos to immediately load.
I am testing this now localy to see who the culprit is. Please hang on.