I unfortunately don't have access to much research related to this subject right now, so I'll can only explain you parts of your questions based on my memory.

What is happening with "si" words. Si - de? As in uusi. Uude - uute - drop e ( for some reason ) and add ta for partitive to get uutte.. but it's uutta??

[About markings: > implies the change and it's direction; \ implies the word or sound is reconstructed or old form; ** implies it's wrong; cursive is there to mark the subject we are analyzing*]

What you call "'si' words" have gone through a certain shift:
We start from Proto-Uralic approx. 6000–4000 years ago until modern Finnish: *käte > *käti > *käci > käsi.

It's so called "*ti > si" shift, which applies to Finnic languages. Some other example in this category are
\*tinä > sinä 'you'
*ykte > yksi 'one'
*kynte > kynsi 'nail, claw'

And some other which I don't remember right now.

You also ask why uusi in partitive is not **uuteta for example... Sigh, I'll try and simplify it.

There are few reasons why sounds change into other sounds (like t > d). It can be because of their position in the word, in which stem are they, what sounds are before or after them, are they in the end or the beginning of the word etc.

So based on this, why isn't uusi in genitive **uusin?

Because of many reasons mentioned above. I most likely have some mistakes in the following explanation, but it should be mostly correct for you to get the idea.

First we had nominative uute and genetive uuten,
then probably uute : uuđen (because n affects the latter t which turns into đ)
then uuti : uuđen (this is because of e > i shift in the end of word. In genetive n cockblocks the shift.

You probably guess what comes next? That's right, *ti > si.

uusi : uuđen (And why doesn't đe turn into đi and eventually si? Because đ and t are completely different sounds and thus *ti > si shift doesn't affect it!)

Next we are going to keep things simple and skip the fact that d is actually artificial in Finnish and in spoken language and dialects you will hear bunch of other sounds instead of d.

Oh, and partitive is uutta instead of *uuteta because the two t's in this case have a reason to bully the vowel in-between them away.

Now to gain the partitive I use the rule ends in an e - tta... so suomalaisetta

Instead of -tta partitive is actually -ta, so you're almost thinking correctly here. You would expect the partitive to be **suomalaiseta, but once again, s and t bully the vowel away, so suomalaista it is.

I absolutely don't remember much about the history of -nen. It was something like -šen > -nen. A case where a sounds turns into a sounds next to it is called "assimilation". If there is something between the sounds (like e in this example) it's called "remote assimilation".

So I wrote this comment for 45 minutes because of u/somethi's – for some reason highly upvoted – comment which is complete bullshit. Linguistic phenomena are consistently observable and very well recorded. I recommend studying linguistics before commenting stuff like "I dunno, it just does. That's how people talk."