Does it take some time until the counter shows the new "share"?
It does look like there is a delay. Honestly, we don't have a definitive answer on precisely how long it takes to show up in the Count. There is no obvious documentation on any of the Developer pages for the Tweet share button.
I can definitely reproduce the behavior, as I tweeted your first article twice. It showed up in my twitter stream, but the count was still zero for about 30 minutes. Then I refreshed your page again and it was 2. I also just tested one of our pages on sourcecoast.com and am having the same issue with it not increasing right away. I'll check soon to make sure it increases. [EDITED TO ADD: our count increased on sourcecoast.com; there was just a delay]
By the way, there are also other posts on the Twitter Developer Forum from within the past two weeks for users having trouble with count increasing right now too. Maybe Twitter has changed something recently causing the delay.
Do i have to wait for facebook to rescan my website?
As I mentioned, you can manually re-scan some of your pages to test (or all if your site is not that big). Or you can wait for Facebook to do it.
The like counts resetting to zero is expected, as the Like action is tied to a specific URL. When you removed the og:url tag, you're essentially setting a new URL that Facebook sees for the article and the count will be for the "new" URL.
I think this should help explain both issues, but let me know if you need more help or clarification.
-Melissa