When you double click on a path in Inkscape, you enter in node editing mode and you can edit the path.
You can return to select mode by clicking the arrow in the toolbar on the left (or by hitting the keyboard shortcut, usually F1):
In select mode you can resize or rotate your shape.
You can change the stroke attributes using the button on the top toolbar.
The behaviour of the stroke while resizing depends from the status of the buttons on the top toolbar (see above and give a look here).
Regarding your file, after the import in Inkscape you see the structure of your image by changing the Display Mode to outline:
The contour is made by the difference of two closed paths, each eye is a closed path, the smile is a closed path and there is an open path (perhaps a test?):
Let's return to Normal mode. Select the object and ungroup it by clicking Shift-Ctrl-G until you got a message of no more groups to ungroup.
Select just the "cloud" (the contour of your face) and change fill and stroke, e.g. with a blue fill and a black stroke of 5 px:
As you see, in this way you have evidence that the "cloud" is not a path with a heavy stroke, but is a composition of an internal and an external path.
In order to have a single path, you must select your "cloud" and Break Apart them with Shift-Ctrl-K:
Now, delete one of the paths (the internal, in my example) and change stroke and fill as desired.
A similar operation should be made on the mouth (you can cut the path and keep an open path). And perhaps to the eyes.
In this way, you can scale the image maintaining the stroke attributes:
In my opinion, such an operation (doodle with a tablet, export and import in a vector graphics editing program) is possible, but you need a bit of work to adapt the result to your specific needs.
It could be simpler if you have in advance an idea of the desired result: you can choose the proper tool from the beginning.