Should store input format in the database? When using the WYSIWYG editor, should I store not only the text but also the format ("Full HTML"/"Filtered HTML"/...) in the DB?
Yes, if you have your own form elements that are using the WYSIWYG, and are storing those results in th database, then you should also save off the input format. I did a survey of well-used modules looking into this a few weeks ago, and didn't find any that didn't do the same. 

Even if you don't need it now, you will have some need in the future tho render out that input, and you will need the format to pass to check_markup.