No, that would be den tolfte sidan. (Yes, it's spelled with an f, it's weird but at least it's phonetically right.)
In my opinion, it is more usual to say sida tolv. I think that the attributes before numbers are used without articles, e.g. plats trettiofem, rad elva. However, the plural forms are specified, e.g. sidorna tolv till femton.
You're wrong about sida, a quick look in Korp gives 16 hits for sidan tolv and none for sida tolv. I checked with some other more common numbers too, and the definite form is overwhelmingly more common. It's not wrong to say sida tolv though.
You're right about plats 35, rad 11, in that case one would not use the definite form.
Who knows, it just seems to me to be a grammatical choice that stuck however long ago.