Doubt regarding cache hit ratios and access time
I think, that the part with some other useless information... in your problem specification must contain some information on solving your problem. As I was trying to calculate the numbers that you got I found some errors, and I thought there must be some more input. Let me first explain:

Your formulas are incorrect for both Method 1 and Method 2. When accessing some memory level you ALWAYS access this level, not with probability! Explanation goes something like this: You ACCESS $L1$, if you MISS (some probability), you go to $L2$. On $L2$ you again make an ACCESS, and if you MISS (with some probability), you move to $L3)... 

So Method 1 with your first problem would be: 
$$[T1]+0.1([2*T1]+0.1*[3*T1])=1.23[T1]$$

And Method 2 for first problem: $$[T1]+0.1([T1+2*T1]+0.1[T1+2∗T1+3∗T1])=1.36[T1]$$

So now looking at your second problem, you can solve it by using Method 1:

New Method 1

Effective access time = 20 + 0.2*(120) = 44ns
Increase by 40ns, so new time = 84ns
84 = 20 + (1-h)*120
Hit ratio = 0.467


Note that here I assumed you made a mistake with answer $(C)$ in your problem description (it is not $0.4467$, but it is $0.467$).

Now if you would have miss penalties defined in your problem statement, then you would use Method 2. Miss penalty just says, that if you do not get a result in your current level, then you have to copy it from the upper level. Now let us assume, that miss penalty on second level is 300ns (that means copying the contents to level 1 and accessing it). Everything else stays the same. The calculations would go:

New Method 2

Effective access time = 20 + 0.2*(120+0.2*300) = 56ns
Increase by 40ns, so new time = 96ns
96 = 20 + (1-h)*(120+(1-h)*300)
Hit ratio = 0.65840


One other caution... You need to use the same formula for calculating the average access time if you chose one method. (In your Method 2 you calculated access time by Method 2 and then calculated the new hit ratio with Method 1)

So how do you know which method to choose? From the problem description! If you have miss penalty given, then you take the second approach. If you do not have miss penalty, then you use the first approach.