1. The plugin says that I have to use https://foursquare.com/oauth/register , but page with API credentials is on https://foursquare.com/developers/apps actually.
You're absolutely correct; Foursquare changed the URL of their app registration page. The next version of the plugin (which is currently being finished and polished) corrects this and refers to the new URL.
2. Foursquare disabled http-pages for pushing, now only https-pages in allowed. Therefore I selected "Disable pushes for this app".
WP Quadratum doesn't support push notifications (in fact, I'm not even sure this is possible via a plugin except via some AJAX cleverness and even then only maybe).
When creating a Foursquare app, the download/welcome page and redirect URI fields are mandatory. Push API Notifications should be set to disabled. All other fields should be left as is.
3. I use https://foursquare.com/developers/apps to create Client ID and Client Secret, then drop them into plugin settings. It is OK, but when I am clicking on "Connect to Foursquare button", I have got a system error 500.
I can't seem to reproduce this ...
I've just gone through a clean (re)install and activation of the plugin on a test WordPress install, that's public facing (needed due to the OAuth callback). I did the following.
1. Uninstalled and deleted WP Quadratum
2. Re-installed and activated the plugin
3. In the plugin's admin settings, entered the Client ID and Client Secret from the Foursquare My Apps page, then clicked on Save Foursquare Settings.
4. When the page reloaded, I clicked on Connect to Foursquare and was able to authenticate via OAuth and my Foursquare credentials.
All worked fine (but see the last part of this post below for another caveat). This might be down to another plugin or theme issue that I've not come across. Have you tried the standard disable all (other) plugins and revert to a stock WordPress theme to see whether this recurs?
One final point, due to a change in the way in which GitHub serves up files via
raw.github.com, Mapstraction, the underlying API that provides the map for WP Quadratum is not currently working. I'm in the process of fixing this and this should be the last hurdle to overcome before releasing the next version of the plugin which fixes this issue as well as several others, including the point you raised in 1) above.
Hope this all helps