Why FED/ECB/whatever raising interest rates is bad for stock markets? (I am aware that this is an assumption - my information could be wrong)
This is because interest rates critically determine price of stocks. For example, using simplistic (but for your question sufficient) Gordon stock price model, the price of stock is:
$$P = \frac{D_0}{i-g}$$
Where $D$ is dividend the stock pays, $i$ interest rate and $g$ growth rate of a dividend.
As you can see increase in $i$ will decrease the stock price.
The intuition for this is that stock is nothing else, just infinite sum of discounted future dividends (which grow at some rate $g$). The higher interest rate is, the lower the present value of money that you recieve in future. For example, if I offer you \$100 in 1 year with interest rate 5%, present value of my offer is $\frac{100}{1.05} \approx 95.24$. If interest rates suddenly change to 10% suddenly the present value of the same offer (getting 100 in one year) is only $\frac{100}{1.1}\approx90.91$.
As you can see interest rate critically determines present value of future cash flows. The Gordon stock pricing model is simplistic but even in more complex models you will see the same relationship, higher interest rates mean lower stock prices - ceteris paribus.
Why is it so bad if the stock market crash? I mean if no body wants shares of companies A,B,C or everyone is in rush to sell shares of companies A,B,C, does it mean that the company should stop operation?
A) Not every stock crash has severe consequences for wider economy. For example, Black Monday (1987) was one of the largest stock crashes in history but the recession it caused was very mild (see discussion of this period in Clarson 2007 or see this Fed history blog).
B) Stock market crash can spill over to the real economy because it impedes investment. Again using simplistic, but for your question sufficient, macroeconomic model of a closed economy (See Blanchard et al Macroeconomics an European Perspective Ch 3-5):
$$Y = C +I + G$$
Where $Y$ is the output/income of the economy, $C$ consumption which we can assume to follow $C=c_0 +c_1(Y-T)$ where $c_0$ is autonomous consumption (consumption that does not depend on income), $c_1$ is marginal propensity to save (must be $0<c_1<1$ as you cannot save more that 100% of your income), $T$ are taxes and $I$ is investment and $G$ government spending (for simplicity assume balanced budget T+G$. We can show that the good market equilibrium will be given by:
$$ Y = \frac{1}{1-c_1} \left( c_0 + I + c_1 T \right)$$
As you can see if $I$ falls because investment spending falls (which could occur in the aftermath of stock market crash - when people are not willing to invest into stock which gives company money for further investments), the output/income of an economy $Y$ will fall as well.
What even more because someones spending is someone else income, there will be multiplier effect, so output might drop by more than the fall in investment itself (since $\frac{1}{1-c_1}>1$).
Does it mean no one would buy the goods/services produced by these companies?
No necessarily, as shown above, fall in investment means some people have less income, so they will spend less. The effect here is mainly indirect. It is not that fall stock market crash would make people desire to buy products less in itself, most people probably do not even know if there is stock market crash unless they hear it in the TV, but when companies invest less then someone's income is reduced and thus naturally their spending declines as well.
This being said, negative news might make people panic and decide to buy less goods and services today because they are worried about future, so there can be direct effect as well.
PS: There is of course also an extra bit of an nuance to everything above, I restricted myself to simple 101 models since you stated you are not an economist. But generally more complex models would tell very similar story.
