If the avatar you made has a dimension of greater than 500x500 and is 1:1, you can just import it into Photoshop, and then re-export it as your preferred dimensions (File > Save for Web). Same thing if your image is less than the preferred dimensions, but the only difference is that you're going to experience quality loss due to upscalling. But you probably already knew that.
For the signature, I don't recommend straight up resizing the avatar to the signature's dimensions. Assuming the avatar you made is 1:1, it's going to look awful if it's resized to a size such as 340x40. I would just resize it to fit the signature's dimensions/size while maintaining its original aspect ratio.
But since it's a gif that you're working with, you're going to need to know the following--
You can merge .gif together and you can convert them into smart objects. You're going to need to open the gif in its own document. Instead of creating a frame animation, create a video timeline. If it's already opened up as a frame animation, just hit the button in the bottom left corner to convert it into a video timeline. After that, group all the layers together. Do the same for the other gifs you want to merge. Once you have all your gifs in their own groups, in their own respective document, create a new document where you want all the gifs to be at once. Click on "create a video timeline" and now drag each group from the other documents into the new one. They should all be animated now. If you want, you can convert each group into a smart object, and now you'll be able to resize them, make edits on them, or do whatever else you normally do.
If there's anything that you need clarifications on or didn't understand, feel free to let me know.