I wonder if anyone else in the forum make drawings only with mouse/trackpad 
I do for more than a decade now (self-taught). And since you mentioned profile picture: my current profile picture, as well as most graphics in
my personal website, are also drawn using mouse as well.
My first recommendation for whomever that would like to use mouse (or by extension, trackpad) for drawing,
is to draw in vector graphics rather than raster (pixel-based) graphics you've been familiar with. When you draw in vector, you're not drawing pixels, but rather creating geometric shapes which could be edited afterward; this frees you from having to nail the shapes precisely in the first time (which you often can't do with mouse); then you can move, adjust, and refine individual parts to your liking later; as many times you want. Great quality can be reached as long as you have patience.
Oh, and have I mentioned yet that file size of vector drawing is also tiny (like, kilobytes), while capable to be enlarged infinitely without ever pixelating?
Inkscape is a well-known art software for making these (1). It is a user-respecting
free(dom) software; and its native text-based SVG file format is
libre standard (2), which is basically viewable in all web browsers since mid-2010s, as well as inspectable/editable in text editor for full transparency and your tinkering needs.
If you need further pointers on how to use Inkscape, ask.
(1) Other well-known ones are Adobe Illustrator, Adobe Flash/Animate, and CorelDraw; but they're all proprietary so I can't recommend them. Also, if you don't want to be
chopped into pieces and sold as feedstock into art-laundering machine, stay away as far from Adobe as possible.
(2) Saving in a file format with published specification means you don't have to worry much if you have to change the software down the line.