Why does lowering stock rate of returns increase stock prices?
In the part 1 there must be some omission. Just by itself a random Fed purchase would not reduce risk premiums. However, a targeted Fed purchase can reduce risk premium as non-trivial part of risk in any firm stems from liquidity risk. Through purchasing assets of illiquid firms Fed can provide them with liquidity that they need and thereby reducing the risk that they fail just due to illiquidity. Also as Fizz points out in his comment this is just one example, there are several other channels that can also lead to risk reduction as well.

The intuition behind this is that the stock prices represent present value of the future expected income streams you get from the stock and at lower interest rate the present value of this income stream becomes higher. To see why this makes sense consider practical example where the stock always pays dividend 10\$ in perpetuity. Then in order to actually calculate whats the present value of getting the stream of 10\$ at 10% you can use the Gordon formula which is just based on calculating the value of infinite sum. In simple Gordon stock pricing formula given as $P=\frac{D}{r-g}$ where $D$ is dividend and $r$ rate of interest and $g$ rate of growth which we can for sake of simplicity ignore and assume its zero so we get $P = \frac{D}{r}$.
At the 10% the infinite sum of 10\$ will be worth 100\$ because
although your are getting the dividend in perpetuity due to time
value of money money in future is less valuable than money now and
becomes less value the more you look into the future and hence this
infinite sum in present value terms becomes smaller and smaller
which allows us to actually calculate finite value even for an infinite stream of money. However, now if the interest rate drops to 1% the present value of those dividend streams becomes higher and the stock will now cost 1000\$ as now the future is discounted less heavily. To put it simply, at lower interest rate the money you receive in future have higher value today.