Hello,
a) the issue 1 (print button not being SEO) and 2 (print point to english even when in czech) may be related;
first step would be to find out, if this is related to JoomSEF directly or whether it does not work even when JoomSEF is off;
what to check:
1) make sure that jFrouter is off when using JoomSEF (if you have version prior to 3.6.0)
2) there should be a lang=xx variable in the link (if it is not, then it is a problem, as the Joomla may not recognize the language correctly - also make sure you have cookies allowed in case the lang is configured to be stored in cookie)
Whether JoomSEF will tranlate also print button link depends on the template and how the print link URL is constructed. If it uses standard index.php?... format, then it should be SEFed normally. However, if e.g. some JS code is used, then there will be no SEO process.
2) as regards the the home menu link issue - the problem may be again related to missing language infromation when poiting to the homepage (you may also want to check Language related settings in JoomSEF configuration as regards this and e.g. make sure that the language variable is always added)