Is it possible to fit this Tioga Mc Groove saddle on my bicycle? See images here.

The bar the new Tioga saddle is mounted on is a bit thicker than the one the original (terrible) saddle is mounted on and won't fit as is.
I can take the old saddle off its bar no problem but it seems like the new saddle's bar can't be removed without removing the clamp.  The nut with a slot for an Allen key (it's actually not, it's slightly bigger than a certain Allen key size) is unbelievably solid; it will not budge, which leads me to believe it's not intended to be opened.
I took out the metal bottom inside of the plastic bar with an Allen key.  Insides there's a removable spring and deep inside (inside the plastic) it looks like there's another thing to be opened with an Allen key but it's hard to tell.  Even if it was, I don't know what kind of tool is able to open something like that down a shaft.
I tried a lot of other pointless things, including heating up an Allen key so it would expand enough to fit properly; I'm pretty desperate at this stage.

Do saddles vary for different size slots (i.e. different bikes)?
Is it possible at all to fit this saddle?  If so, how can it be done?
Yes, your bike can be fitted with pretty much any saddle. 

The (metric) allen key on your new saddle is a 6mm. If you're using Standard Allen wrenches its probably a little larger than 7/32". 

The seat post needs to fit your frame. So you will need to mount the new saddle on your old post, or buy a new post that exactly matches the size from your original one. 

It does sound like you've disassembled the guts of your suspension seat post, so if you can't get it back together, you're probably looking at a new seat post. 

There should be no problem mounting that seat on any post that fits your frame. 

I hope that helps.