* The mobile company does not have access to your data. They just issue a special SIM card that stores your private keys. Unlike the article states, there is actually also a small piece of software installed on your phone that is later used for authentication and signing. The way it works is that you go to a site where you want to log in (I'll use Estonian government portal https://www.eesti.ee/eng/ as an example here), you press Enter (or "Log in" or whatever the button is called on the site) and choose "Login with mobile-ID". In Estonian government portal both your phone number and personal code are asked as "usernames" but some sites might ask either one of them (depends on the site's owner). Then after a few seconds your phone will display a screen showing that a connection is being made. At the same time you will be shown a "control code" on the browser screen that you will have to confirm on your phone. This is done in order to prevent some forms of man-in-the-middle attacks. Then you're prompted to enter a mobile ID PIN1 code (used for authentication) into the phone. If PIN1 is correct, the screen on the phone disappears and the website is automatically reloaded with a logged in screen. You can cancel the authentication procedure at any moment. PIN2 is used for signing and works the same.
* Obviously it would take a lot of time to answer all the problems posed in the Wikipedia. I would disagree, however, that Estonian e-elections are easy to tamper. As Estonia is on the forefront of e-voting in the world, all these problems are addressed and analysed in great detail by our Electronic Voting Committee (http://vvk.ee/general-info/electronic-voting-committee/) that is involving leading e-voting experts in their work. There have been many debates about e-voting and rulings by our Supreme Court. The process of e-voting is very transparent (you can start reading about it here: http://vvk.ee/voting-methods-in-estonia/engindex/), it's heavily audited before, during and after the voting, the software is open-sourced (https://github.com/vvk-ehk/evalimine) and very well documented. But to give a very high-level answer to your question about recounting the votes - there are many different applications and servers used in the e-voting process that keep traceable logs. If recounting is needed then these log files are used in order to determine if votes were tampered in some constituency.