Hi,
here some quick answers.
But at first the usual question "Which version are you going to use?"
We still recommend version 3.x.
1. Participants can print out (after giving all answers) their response or download a PDF of it. (i only found a plugin where the admin can print it)
This is standard behaviour with this setting.
2. Send automated reminders to those participants who have not completed/done their survey yet. (there are plugins for that if I am right)
Not automated, you have to start it manually.
3. Participants can save and finish later
This is standard behaviour with this setting
4. Participants have to log in, in order to take part. (is it possible to sync a user database between LimeSurvey and the website?) so already logged in users on the website don't have to login again to do the survey? (actually, each user got his profile page on the website and it would be nice if he got his survey in it)
If this is to understand as "They have to enter a token/password to enter the survey", yes, if you use the tokens of your database as tokens in LimeSurvey and add it to the link.
5. Embed it into the Website (iFrame)
6. (backend) possible to filter users by if they have completed the survey or not?
Standard
7. (backend) order/add/tag participants in groups? So you can filter users by a certain group.
Each participant may have several additional attributes.