If your machine is compromised for example, let's say your BIOS is infected, or an attacker with physical access to your machine installed a a nefarious software then yes, such information may be leaked.
But in practice, as for the precise information you wanted to know through the two bullets you highlighted, I have not heard of that before and I may say no -but not sure- AFAIK regarding how the whole system is functioning especially that Tails benefits from continuous audits and you can run this OS directly from a USB stick.
Apart from vulnerabilities that compromise user's anonymity such as disclosing the user's real IP addresses as it was the case with this vulnerability with Tails 1.1 version, there are two information you may need to know and that may answer partly the questions you raised (Does Tails collect information about its users?):
When Tails starts, two HTTPS requests are made automatically to our
website through Tor:
- A security check is performed to know if security issues have been announced for this version of Tails. The language of the working
session is passed along with this request to display the notification
in the preferred language of the user.
- Tails Upgrader checks for newer versions. The version of the running Tails is passed along with this request.
We believe it is important to notify the user of known security issues
and newer versions. We calculate statistics based on the security
check to know how many times Tails has been started and connected to
Tor. Those statistics are published in our monthly reports.
While Tails is recommended by Edward Snowden for anonymity and privacy concerns, it is not a foolproof system. You can find here a complete list of reported Tails security vulnerabilities.
While Tails is may be the best OS for privacy/anonymity concerns, and you can trust it more than other operating systems for this purpose, I would love to quote you this from Silver Bullets and Fairy Tails:
We publicized the fact that we’ve discovered these issues for a very
simple reason: no user should put full trust into any particular
security solution. By bringing to light the fact that we have found
verifiable flaws in such a widely trusted piece of code, we hope to
remind the Tails userbase that no software is infallible. Even when
the issues we’ve found are fixed by the Tails team, the community
should keep in mind that there are most certainly other flaws still
present and likely known to others.
And remember that not all Tails security vulnerabilities are reported as there are those who find them but do not disclose them and prefer to make business with them; Exodus (from where I quoted the text above) itself sells 0-day vulnerabilities.
Note that a Black Hat security conference canceled the talk You Don’t Have to be the NSA to Break Tor: De-anonymizing Users on a Budget where the presenters intended to explain how, with a budget of $3000, hundreds of thousands of Tor users could be de-anonymized.