Bond ETF vs stocks versus interest rates

The simple answer is that none of those statements are guarantees, they're simply rules of thumb.

• stocks move in opposite direction of interest rates

This is not always true, for example right now longer term (10-year) interest rates have been creeping up because the market expects some degree of inflation down the road, and yet stocks as measured by the S&P have also been creeping upward. If the govt. were to pass a massive infrastructure bill then that could also spur short term interest rates upward if the Fed decided to raise rates on inflation expectations, but all that infrastructure spending could be expected to boost corporate profits which could be argued as a reason to buy stocks.

• bond ETF prices move in opposite directions of interest rates

Often but not always true and greatly depends on what type of bonds (ie, long/short term; investment/junk credit, corporate/government). For example, if short/medium term interest rates are rising, investors might move into long term bonds if they fear a recession is around the corner which would push the price of such bond ETFs upward. So with rising short term Fed interest rates the bond ETFs tracking short term govt. treasuries would be falling (so that yield would go up) but the bond ETFs tracking longer term treasuries and even corporates could actually be rising (meaning the yield on them would be going down). This is actually common late in economic cycles and is known as an inverted yield curve.

• bond ETF move in opposite direction of stocks

Of the three statements, this one is probably the least guaranteed. When COVID first hit in March last year, everything got sold off including both stocks and bonds. When the Fed lowered rates to 0 in response, it created a bunch of cheap money and low borrowing costs which means investors agressively borrowed and that money has flowed into both stocks and bond ETFs.

Finally, to get to your example, if short term Fed interest rates are rising, then that may either signal an impending recession or maybe it just makes cash finally produce an adequate return in which case investors sell stocks and park the proceeds in cash. Meanwhile, short term treasury bond ETFs would need to fall in price so that the yield would rise to approx. match the Fed.