Yes. The logo you design is automatically saved, and you can always download your logo(s) from My Designs
Any browser that supports HTML5, nowadays every modern browser supports, we recommend latest versions of Google Chrome, Mozilla Firefox and Internet Explorer (atleast ver 10).
No - Once you've designed your logo and downloaded files to use it, we can't refund.
Almost certainly. If your logo design has a unique combination of an icon, text, company name, spacing, and colors, then it is very likely you can trademark your logo for your industry. The icon used in that logo will always remain our (Designimo.com) property and it will remain in our database for future use.