When you enter one of the three micro-states, the Schengen 90 day clock pauses during your time there. It resumes when you exit the micro-state and reenter the Schengen zone.
Also, you cannot 'refresh' your Schengen clock in the micro-states because they have local rules that act to prevent it. In San Marino for example, foreigners staying longer than 30 days must apply for a residence permit and these are not given to visitors.
But indeed, each day in a micro-state is a day out of the zone and the 90/180 rule is affected accordingly.
Problems will arise when the visitor ultimately leaves the zone and must undergo a Schengen exit inspection. At this point the person may be asked to prove that they are not an overstayer and the border guard will be entitled to see hard evidence in the form of official documentation. There was a case where an individual attempted to show that his mobile phone registered to a tower outside of the zone and this failed obviously: radio waves have no respect for nation borders but also one could mail the SIM card to a friend in a far away land, put in a phone where it registers to the tower and then mail it back. Apparently hotel receipts can be too easily forged to be successful as evidence.
What the border guard will want to see is a passport stamp from the local constabulary. But the constabulary is under no obligation to issue one, so success is down to personal impact and articulation skills.
Source: Question put directly to Elspeth Guild 6 June 2016, (DT 1568 Understanding Schengen Visas).
Also, according to the article, Andorra seems to do have immigration
borders... So, for example, if I am a US citizen, will going to
Andorra, stop the Schengen visit
Per the above, yes if you go to Andorra the Schengen clock is paused.
(meaning there will be an exit stamp on your passport)?
In the normal case you will have no exit stamp and be unable to provide satisfactory evidence of your time in and out of the zone. As explained you can approach the local constabulary and TRY to get the entry and exit stamps you need.