The second derivative test doesn't always work. If the second derivative is positive then there's a relative maximum at the critical point. If the second derivative is negative then there is a relative minimum at the critical point. If the second derivative is zero then you don't know what the critical point is. It can be a maximum a minimum or even an inflection point. You would have to use some other method to determine what type of point this is (like the first derivative test).
When the second derivative is easy to find and calculate then it may be easier and faster to do the second derivative test instead of the first. Of course, as stated above, if you get a zero then you wind up doing more work.
There may also be times where the first derivative test isn't feasible to use. Suppose you are able to find one critical point (where the first derivative is zero) but it is too difficult to find all of the critical points and you just want to determine what that one critical point is. If you don't know where the other zeros of the first derivative are you can't be sure what to pick for a test point. Even if you pick a point really close it's possible that it might not be in the interval between the known zero and the next one (another zero might be really close too). In this case you'd have to use the second derivative test and hope it isn't zero.
The method for finding inflection points is very similar to finding relative extrema.
Find all the zeros of the second derivative. These will be your critical points. Not all zeros will be inflection points so you need to test whether they are or not. The sign of the second derivative tells you which direction the concavity is. Examine the intervals between zeros to the left and right of a particular critical point. If the sign of the second derivative is defferent on each side of the critical point then it's an inflection point (since the concavity changes). Alternatively you can use the third derivative to determine if it's an inflection point. If the third derivative isn't zero then it's an inflection point. If the third derivative is zero then you need to use another method to figure it out. See how similar this is to finding extremums?
Yes you can.
Let x=a be the value of x for which f'(x)=0. This says that f(x) has an horizontal tangent at x=a.
Now, look at the sign of f'(x) for x<a and x>a.
If f'(x<a) <0 and f'(x>a)>0 then the function f(x) DECREASES for x<a and Increases or x>a => x at x=a the function has a minimum or a horizontal inflection point.
You should work out the other cases
No you cannot find the relative maximum or minimum with First derivative test in case of a polynomial function. Second derivative test is essental in such cases.