I am actually a fraud, as I only remembered the existence of Paper mario after i saw your profile picture when I was searching for a new avi. Then i remembered that Dimentio is a pretty cool guy, and also the latest boss i battled, and he was just perfect for a profile picture.
Sorry for my lack of knowledge, but how does one make a transparent background? (I blame the fact that I have never needed a transparent background.) (I also need a different picture, as the foot of my Dimentio is below the border, and there is lot of empty space on the top.)
Also, f-777 - dance of violins is a good song.
I am assuming someone did that for you (it's not complicated for this one), but all of the problems you have have simple solution once you learn them, so let me tell what I know
The very first thing you can do is simply getting the best version of an image. Say you google search it, there;s a feature to search images similar to a result, it's certainly not perfect, but it works. There coudl be a .png version with transparency high resolution of the image you want which is going to make your life a lot easier afterwards. Try to avoid .jpg because the compression is often quite annoying, doesn;t support transparency and if compressed enough, there would be too much artifact to actually try to remove the background. PNG supports transparency, has much less compression and jsut looks good with the only downside being more filesize......but tbh.....you need a lot to lag something.
But most of the images you will find will not fit if you need standards such as the avi here. They are 150x150 unless you are premium which gives more height. So ideally, you would want the exact size, but it;s not really possible so the trick is simply to use the image editing program of your choice and do some tricks.
For the program to use, there;s mainly 3, photoshop, paint.net and gimp. The first is really good, btu costs. The second is free, but it;s windows only. The last one is open source, multiplatform and free, but it has its quirks. There;s paint.....but tbh, don;t use it, jsut paint.net is a lot better.
They all work the same way however. To remove a background, you have to make a selection and simply delete it. making the right selection is the tricky part, you can select shapes like rectangle and stuff, but the easiest one is I think people call it fuzzy selection or something, but it;s jsut a pathfinder that checks similar colors. You can adjust the threshold so ideally, you want an image that hgas a cleanly defined background that only has one color and the edges of the subject to be contrasting that color. This is exactly what you had, a white background with the subject having black lines and edges (most Paper Mario characters are so easy to do this becuase of this).
The problem is this is the ideal scenario, soemtime it;s tricky to do this either becuase of quality, colors etc......it;s soemtime hard to do and it woudl leave some mark on the image, but if you can find a better image, that might work then.
The second problem you have is even easier to solve. Say if you have an image that is more in an horizontal position and you want it to be a sqaure, well you could simpyl resize the canvas to be square. All it does is it adds dummy spaces, but your subject is unchanged, It does appear the subject is smaller, but it;s MUCH better than having it cropped.
And the last thing that is really basic is scalling, it;s not wrong to scale down if it is preserving the proportion of the image, all the image editing I used has a way to scale down proportionally withotu using stuff like a calculator. You DO loose quality, but you HAVE to do this and a proportional scale down is actually harder to notice if it;s not too much. So, the best case is you have a super high res image that you scale down small, it had so much data at the beginning it might be very hard to notice it.
I explained mainly how you do it, but it varies for each tool. I can teach these in gimp, it;s very easy, but you might jsut want to google it because like I said, once you know how to do it, it;s very simple.