How To Create a Remote Views Search Form - D7
I would suggest writing your own form in a block which you can place on the home page. I don't see anything wrong with writing the form in HTML - though if you generate your form the Drupal way using the Form API then you'll ensure you get consistent themeing across your site, you'll be protected against XSS attacks and so on. How to use the Form API is beyond the scope of a single question ;
There is a module that allows you to use POST queries with views exposed filters : http://drupal.org/project/views_expost - hopefully this should deal with your long URL problem.