Would rising interest rates (as set by fiscal and monetary policy) increase the profitability of companies that lend out money, and thus increase their equity value?
Such a question is not directly answerable because there are many factors that go into a bank's profitability, but here's something to consider:
Bank's don't make more profit just because the federal interest rate is higher. Banks must pay interest as well, either to entities that have large deposits with the bank, or to the fed as they borrow money to lend out. So a higher federal interest rate alone is not enough to automatically make more profit for the bank. They much make more money in interest and other fees than they spend in interest and other expenses.
It's possible that higher risk-free (federal) rates will mean banks charge a proportionally higher spread, but higher rates also means less borrowing (all else being equal) so there are several factors in play.
To be a bit more concrete, I looked at the history of the 10-year fed rate and a "financials sector" ETF (XLF) over the last 22 years (the age of the ETF), and the correlation of returns was only 0.296, with an r-squared of 0.088. Meaning that interest rate changes are not highly correlated with financial sector returns, and only account for 8.8% of the returns of the ETF. So a rising interest rate might help but it's not a large factor.
So the answer is "more likely yes than no", but it's by no means guaranteed.
Rising interest rates suggests an economy that is strengthening. An improving economy means increased employment which improves consumption, making companies more profitable, driving the stock market higher.
Banks earn more from the spread between interest paid on deposits and lending money at higher long-term rates. Insurance stocks also fare better in higher rate environments. Due to increased consumption, discretionary stocks also benefit. However, higher rates isn't a benefit across the board. Higher borrow costs reduces corporate earnings and may reduce expansion if rates get high enough.
So yes, higher interest rates can be beneficial to the stock market but at some point, if rates get high enough to stifle the economy, growth and profits slow and many investors move out of the stock market into Treasury bonds.
Generally - yes. Banks make money on the spread between what they lend out and where they refinance - and a 20% spread is smaller on smaller interest rate (i.e. 1% to 1.2% is 0.2%, 2% to 2.4% is 0.4%). And yes, financial institutions generally refinance. Lots of financial institutions are in serious trouble because of the low interest rate, which also makes deposits unattractive for investors. This obviously may be offset by lower volume, but - in many countries banks (which mostly do lending, not the investment stuff that is in investment banks in the USA) are basically on life support ever since the financial crisis for this reason. OTOH higher interest rate means - well, your loans may default (particularly old loans - companies tend to go bankrupt when paying a lot of interest rate).