1. Speculation / belief
2. Approx 10x less influential is trade, most currency transactions both by volume and by value are in speculation which is underpinned and driven by beliefs in a particular currency. Some estimates put trading as low as 2.5% of all forex. Trading does impact the price of a currency, but in miniscule proportions to speculation
3. We can fairly confidently say tax collection is a lower influence than trade because of so many examples (and extraordinarily rare counter examples) but I don’t know of any reasonable way to quantify how much less the effect is. The threat of incarceration and / or asset seizure drives demand for currency which influences its value.
Places like Argentina are fascinating (said while acknowledging the extreme hardship being placed on people who have to use Argentinian currency!). In recent history all of the above have had visible influence on their currency along with pegging / dollarization which attempts to introduce a control on all the above factors with various degrees of success.
Simple: demand.
The value of Crypto products is based on scarcity dictated by the computationally intensive nature of certain algorithms, and on people believing those products have value.
You mean math?