The vast majority of voting systems, almost all of them as deployed in
democratic systems, do not consider voter stability.
Actually, the vast majority do take voter instability into account. A little less than half the countries in the world (i.e. more than half of the democratic countries) use a Bicameral system where the two chambers of parliament are elected in a different way or at different times and both are needed to change laws. That setup is expressly designed to prevent major changes after a single election win by a party. Other systems have protections like a president's veto power or judicial reviews of laws.
A second way that voting systems account for voter (in)stability is by not having the option of a referendum at all, or a non-binding "advisory" one at best. Those result in some extreme political contortions as the government finds a way to work around the unwanted vote outcome or redo the vote, but very rarely result in real change.
A prime example is the attempt to introduce a European Constitution. It was rejected by referendum in France and the Netherlands and supposedly abandoned. In practice, most of the big changes proposed ended up in the Treaty of Lisbon word for word, which conveniently did not require a referendum in France or the Netherlands. That treaty did encounter another bump in the road, being rejected by the Irish in a referendum, but a repeat of the referendum was organised and then the 'Yes' vote won.
So, the answer to your Q1 and Q2 can be found in the structure of governments the world over. The people can vote, but they cannot force the elected officials to take any action or make any law. Officials or parties elected in a wave of emotion or discontent cannot make big changes right away, they need to last until they can get members elected to the other chamber of the parliament a few years later, by which time voters usually have come to their senses again.
For your Q3, voter stability is hard to define. I would argue it needs to include informed support for a platform, program or position rather than an expression of protest or discontent. I.e. a voter's preference is likely to be more stable when they say "That is where I want the country to go" as opposed to "I'm unhappy with where the country is going, it needs to change!".
By this (incomplete) definition, the vote in the UK would be considered extremely unstable, as there was no plan or platform available for "Leave" voters to support, only promises without any basis in fact. The "Remain" campaign did not do much better, mostly outlining why a "Leave" vote would be bad.