Q1: Is the data correct(are they really uncorrelated)?
What sort of google scholar search did you do? Generally research suggests there is a relationship between interest rates and stock returns, so your search was not correct/representative.
Citing from recent cross-country study published in decent journal by Assefa et al (2017) that analyzed the relationship between interest rates and stock returns around the world [emphasis mine]:
Using dynamic panels, we find statistically significant negative effects of interest rates on stock returns in the developed countries, consistent with the expected cash flow hypothesis. In the developing markets, however, the world market portfolio is the sole determinant of stock returns. The contrasting effect of interest rates change on stock returns can be partially attributed to differing monetary policies and to the more mature capital markets inherent in developed economies.
This is by no means only study that finds there is correlation between the two. There might be studies that show they are weakly related in some special periods but over long term and cross-country there is clear significant relationship.
Q2: Are there any consensus, or at least a good theory, on why this happens?
As explained above there is generally consensus that interest rate and stock returns are related. Given the answer in 1 this is moot point.
Q3: Real life implications - as a retail investor with pretty strong faith in EMH, is it rational for me to move my money from stock market to bank account because the interest rate went up?
This site is not for investment advice any financial advice here is strictly off topic. This being said I suppose the question can be slightly change as asking what a rational agent within a model would do which would be on topic.
Within economic model of a rational risk-neutral representative agent if there are two different assets (e.g. stock $A_s$ with expected return $r_s$ and savings account $A_a$ with expected return $r_a$) then all else equal if $r_a> r_s$ the rational agent would only use $A_a$ for saving and if $r_a < r_s$ they would only use $A_s$ (as long as $r_a$ and $r_s$ are fixed as endogenizing them would lead to the closure of such arbitrage opportunity).
If a rational agent is risk averse and $r_a> r_s$ they would still only use $A_a$ (since saving account should be less risky), and if $r_a < r_s$ an agent might want to get some stocks depending on the parameters (risk aversion, by how much more $r_s$ exceeds $r_a$).
If a rational agent is risk loving and $r_a> r_s$ agent might still prefer some stocks (depending on coefficient of risk aversion and relative sizes of $r$). If $r_a<r_s$ a risk loving rational agent would always just get the stock.