Well, if you're just looking to fool around, GIMP is free and can do both the first and last options. I've never made a gif using a video with GIMP, but I know it can be done.
As far as the other one, the only thing I've really used with LoL is OBS and it seemed to work fine for me.
As far as the other one, the only thing I've really used with LoL is OBS and it seemed to work fine for me.
Just about all of my sigs I have made using gimp, and gifs are possible if your experienced in layers.

Example of a gif signature I made in gimp
For recording you could try the xSplit GameCaster there are free and paid versions of it, but I have also heard that OBS works great it's harder because the interface is less user friendly, but thats just from what I have heard I have never tried using OBS.
For a Video Editor you could use Magix Movie Edit pro I have 14, but 16 is available from steam for around 100 bucks, and it's what I mainly use, but Sony Movie Edit pro is a lot better and it's cheaper I use both for different things.
A few gifs I made with Magix
they are awful
Hopefully all that helps out but when it comes to buying an editng program don't just take recommendations do your homework thoroughly (^_^")
but if you are looking for free editing software you can use Blender it's CGI animation program but it does have a full blown built in video editor that's 100% free.

Example of a gif signature I made in gimp
For recording you could try the xSplit GameCaster there are free and paid versions of it, but I have also heard that OBS works great it's harder because the interface is less user friendly, but thats just from what I have heard I have never tried using OBS.
For a Video Editor you could use Magix Movie Edit pro I have 14, but 16 is available from steam for around 100 bucks, and it's what I mainly use, but Sony Movie Edit pro is a lot better and it's cheaper I use both for different things.
A few gifs I made with Magix

Hopefully all that helps out but when it comes to buying an editng program don't just take recommendations do your homework thoroughly (^_^")
but if you are looking for free editing software you can use Blender it's CGI animation program but it does have a full blown built in video editor that's 100% free.

Sig by The_Nameless_Bard
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Thanks in advance!