Mike, 1) Theoretically, increasing during Risk Off and 2) Yes and no. Sorry, it’s not black and white.
If the market tone is viewed as Risk On this means investors are prepared to “gamble” more so they may pile into Tech stocks, Bio Stocks or the S&P etc (often with leverage) where the general risk is higher but the returns are also higher.
Risk Off on the other hand is where investors are more risk averse due to various reasons such as Fed monetary policy, a terrorist attack, natural disaster, black swan event etc so they want to take their money out of risky Tech/Bio/S&P stocks and instead place funds in assets which are deemed less risky, but generally speaking with a lower rate of return. These asset classes could be Gold, Yen, Dollar, Swiss Franc, Bonds etc, so theoretically, if more investors are buying “safe” assets such as Gold/Yen/Dollar/Swiss Franc, they will rise in value.
Oh and if anyone ever tells you that the market is Risk On (S&P should rise) so it means USD has to go down, ignore them. They often rally together, fall together, but sometimes also have an inverse relationship.
This type of sentiment comes and goes. View it as rule of thumb that sometimes happens rather than a hard and fast rule.
Risk On one day to Risk Off the next day is fairly normal. It all depends on what happens around the world with regards to geo-political events, high impact economic news or something as simple as it being an Amazon prime day (believe it or not). For instance, Amazon announce 2 days of heavily discounted products, investors believe this will increase Amazon’s share price so they invest in Amazon and perhaps other FANNG stocks as they often rally together. This generally means these investors will pull their money from safe-haven assets such as gold and bonds so they can take advantage of the likely S&P rally. This is Risk On. But then, the day after the Amazon sale, Turkey bomb a neighboring country for whatever reason which sparks concern across the world as political leaders come out and condemn the actions of Turkey. This makes investors risk averse due to the uncertainty so they may take all their profits from the Amazon/S&P rally and perhaps even the majority of their entire capital and put it in US or German 10Y bonds (or currencies like Yen, USD, CHF) as they have the highest interest rate at that particular time, until things calm down. This market would be viewed as Risk Off.
Hope that helps mate.