Filter and crop an image
First, let's start with the wrong part of your post:


  to apply fft, images must have dimensions which are the multiple of power of 2.


This is incorrect. FFTs do not have to be a power of 2.  The only requirements for applying an FFT are that you can decompose the signal length into its prime factors and that you have an FFT that can deal with these prime factors.


  to apply the fft, the dimensions of the image and the mask must be equal.


This is correct. To do the point-wise multiplication in the frequency domain, it's easier to do when both image and kernel FFTs are the same size.


  we can take the max of the width and heights of both image and the mask, round them to next power of 2 and, pad them.


The power of 2, as pointed out before, is incorrect.

the width and heights of both image and the mask, round them is also incorrect.

Your second option:


  we can take the sum of the widths and heights of the image and mask, round them to next power of 2 and, pad them.


is closer to the truth, but still incorrect.

Suppose your image is $N\times M$ and your kernel is $K \times L$. To use the FFT to do linear convolution you need pad both of them to be $(N + K - 1) \times (M + L - 1)$. If you don't you'll get time aliasing from the circular convolution.

Now, to answer your questions:


  So, in both of the cases we are actually obtaining oversized output images with padding. Am I right?


Yes, to avoid circular convolution results (and the associated "time" / "spatial" aliasing), you need to pad.


  Suppose, I pad an image and get it FFT-filtered (Sharpened/Blurred/... and so on). So, I obtain a filtered image with padding. Now, I need to remove the padding to obtain the filtered image without padding. Am I correct?


Yes. If you're interested in an in-place replacement (so that the resulting image aligns with the original unfiltered image), then you need to remove the padding.


  If YES, should I just crop the image? Or, is there any better way possible altogether from the ground up?


Yes, just crop the image at the right place.