"I would like to ask if there is a way to only have the homepage available in several languages. (To display the language changer only there and when navigating through pages to be back in, say, english)
Yes, you can place the language changer only on homepage and this will allow you to switch to and from any available language. You will also need to implement few lines of code in your e.g. BasePage to switch back to e.g. English for any other page.
could you give us a tip on how to create a dropdown menu instread of the flag images?
You can build your own simple WebPart with drop down and these few lines of code to do the binding:
/// Bind language.
MonoXCacheManager cacheManager = MonoXCacheManager.GetInstance(CacheKeys.Modules.LanguageChanger,
IEntityCollection2 dataSrc = cacheManager.Get<IEntityCollection2>();
dataSrc = LanguageEntity.GetLookup().Entities;
languageChooser.DataSource = dataSrc;