Hi Harri - sounds like you've already got quite a lot of advice!

Here's a bit more that I'm not sure was mentioned directly above.

Right now it looks like you've got the outline on to the sphere how you want it like this:



So you can now cut up the sphere to give you just a teddy shaped sphere surface piece by using the Edit > Trim command. Select the sphere, run Edit > Trim, and select your curves there as the cutting objects. Then the next stage in the trim command is to pick which pieces you want to discard - in this case you would pick these 3 pieces to discard (or you can switch the mode to "keep" where you pick which pieces to keep instead and pick the one big piece you want to keep):



Then when you finish the trim you will be left with a surface that is a piece of a sphere that has your teddy outline for its boundary like this:



Now if I understand you correctly that is one part of the shape that you want, but you want to punch it out to thicken it but in such a way that it follows the surface normal and stays perpendicular to that surface.

That kind of "perpendicular to the surface at every point" type thing is called a surface offset, and there is a tool called Shell which can be used to thicken a surface into a solid piece by generating an offset surface and then connecting the edges between the original and the offset. See here for a couple of examples:
http://moi3d.com/2.0/docs/moi_command_reference7.htm#shell

So at this point you select that sphere surface piece, and then run Construct > Offset > Shell (it's the second grouped command that pops out when you click on Offset), and type in your thickness that you want and you will get this result:



Here it is shown at a different angle:



You can see there that the inside piece becomes a second sphere at a smaller radius - that smaller radius sphere is a constant thickness away from your original sphere piece:



Generating a surface offset like that gives a different kind of result than extrusion - you can also extrude a surface as well but it will make this style of shape instead:




So basically to get a "constant thickness and perpendicular" result you will be wanting to use an offset type tool to do that, and the Shell tool is probably the most convenient for your case since it will thicken a surface fragment into a solid generating the side wall pieces at the same time as well.

Shelling for thickening a surface tends to work best when the thing you're trying to thicken is all one smooth piece.

Hope this helps give you another way to get it done!

- Michael