1. Why is the post reward amount higher before the payout than after?

Answer: The post reward shown before the payout includes the entire amount (author + curators) as opposed to after the Payout point when it only shows the amount paid to the author.
To get the full info, head over to steemd and checkout this section:

total_payout_value is what the post's reward amount will show after the Payout and what goes to the author.
curator_payout_value is what was paid to the people who upvoted the post, the curators

2. If a comment on my post gets a big reward, do I get anything as the author of the post?

Answer: Unfortunately, no. There was initially a 50/50 split but has been changed to 100% for the author of the comment. Be sure to upvote that good comment early so that you get some curation reward

3. Do comments upvoting actions also have the penalty in the first 30 minutes like the posts? (proportional amount is moved to the author depending on the moment of the upvote for the first 30 minutes - e.g. 50% reward at 15 minutes after the post is created, 0% after 30 minutes.)

Answer: No.

4. Does downvoting bring any reward to the curator?

Answer: No, but it helps the community greatly if you curate the spam posts, trolls, bad bots etc.

5. Does downvoting use up my Voting Power?

Answer: Yes.

6. Does authoring new posts or comments use up my Voting Power?

Answer:* No.

7. When is my Steem Power taken into consideration for the post or curation reward, at the time of the action (new post, upvote, comment) or at the time of the Payout?

If I have more SP when the Payout takes place than when I posted the comment, does that mean the reward is higher?
Answer: At the time the action takes place (e.g. When you author a post/comment). No, Powering Up in the meantime won't affect the reward of a previously submitted post.

8. When can I start to Power Down?

Answer: If you created your account and received a sign up bonus (the 3 Steem(, you will need to Power Up at least 10x what you received before you can power down

9. Where can I see who I follow and who follows me?

Answer: A built in option is in the words from what I understand but right now the best place would be on @jesta's https://steemstats.com/. Enter your username on the top left and on the first tab, Activity, you have 2 subtabs with the activity of your followed accounts, Recent Activity, and the list of the people you follow:

10. Why aren't the exact timestamps mentioned on posts or comments?

Answer: They are, just hover the clock icon for a post or the x hours ago on the comments and you'll get the exact timestamp.

11. [Fun one]: What would be the reward of a post if everyone would vote only for that post on that day (author + curators)?

Answer: 28,800 Steem ( split 50% in Steem Power and 50% in Steem Dollars)
The actual reward for the author/curators depends on the time the upvotes are posted as if earlier than 30 minutes then some of the reward of the curators goes back to the author.

EDIT:

12. [BONUS ONE] Where can I see what new platform improvements are being worked on?

Answer: Check out https://test.steemit.com. Login with your standard Username and Password and check them out. These may not be all but some in the final stage will show up nicely.

I will continue to post new questions and answers as both come up. A thanks to @pharesim for answering some of them!!

Please let me know if any of the answers above are incorrect or if you have any other questions that might need answering.