<p>So, I'm having this problem where almost anything I print with a section(s) that is not directly connected to something below it breaks when I try to pull the small filaments meant to hold it up during printing off. For example, I 3D-printed a Rayquaza(<a href="https://www.myminifactory.com/object/rayquaza-23624" rel="noreferrer">this one</a>) from Pokemon for my little brother, and as I was carefully pulling the filament from under the mouth, the whole head just snapped off. Does someone have a recommendation as to a way to get the small filament off without breaking the object? Would a solution just be to print it bigger and see if it holds up better, or is there something else I can do? Thanks.</p>

The small filaments you remove that hold the parts up are called supports. The one model I located on Thingiverse clearly requires a number of supports, as the model is not easily designed for 3d printing with FDM printers. It would be better printed with SLS, but that's not the focus of your question.
You don't specify how large you printed the model, but certainly a scaled-up version will be stronger at the weak points. You will want to use sharp non-shearing cutters to clear away as much of the supports as possible, without torquing on the model.
Another option which also reduces the forces on the model body is to use a soldering iron to smooth and clear/cut the supports. If you are able to use cutters and not damage the model, the soldering iron can remove and flatten the remnants of those supports.
Please note that if your careful work has resulted in a model that snaps to pieces, your little brother will soon destroy the successfully cleaned up model just as easily.
If you have skill with 3d modeling software (Meshmixer and Blender come to mind for such organic models), you can add insignificant items to the model to provide functional support. Would the Rayquaza look fiercer if you 3d printed a cage as an integrated part of the model, using the bars of the cage to provide support?
I successfully printed a model that was created by an artist unfamiliar with 3d printing restrictions. The support material was wash-away PVA. I provided the model to the "owner" who washed away the support material and snapped the legs in two. It's sometimes impossible to solve poor designs. You have a good chance if you build a cage for this one.