Fill factor only comes into play when an index is created or rebuilt. It is the amount of consumption for the index of the leaf level pages that are filled during these operations. (see the note below for more clarification on affected page levels)
When there is a DML command to data (INSERT, UPDATE, and/or DELETE), it will happen to the corresponding affected indexes. In other words, if you have a page that is 20% filled and you insert data into that page, the page will contain more than 20% of the data (let's say 35% just for example's sake). Do another insert, now the page is 64% filled. Rebuild the index, and the leaf level pages will now relatively contain the space percentage that you specify (or implicitly the default value for the server).
(Note, when you don't specify PAD_INDEX to be ON, fill factor is only applied to leaf level pages. But when you set PAD_INDEX = ON, the fill factor will be taken into account for intermediate-level pages of the index. The default is OFF)
The reason to adjust fill factor (instead of using the default 100/0) is so that you minimize page splits when inserting or updating data. But keep in mind, nothing is for free. The lower the fill factor, the more space data would normally take up. If you keep an 80% free page space for your indexes, they will consume a relatively larger amount of disk space, which can lead to more reads.
From my understanding, when the data is inserted there will be roughly 20% of data in the pages. When the data is updated, however, it will expand to more than 20% of the index, upto filling it and generating a page split, right?
When data is inserted it will insert into the appropriate indexes at the appropriate page. This could and most likely would very well cause the page consumption to be higher than the fill factor.
A page split will happen when new data is added to a full index page. SQL Server will then split the page and approximately place half of the data from the full page into a new page. Again, fill factor does not come into play here.
A legitimate reason to lower fill factor would be to minimize page splits, therefore minimizing index page fragmentation.
PAD_INDEX only applies to non leave level pages. By default those are filled full.
So specifying PAD_INDEX=ON together with FILLFACTOR is only useful when you specify a fillfactor smaller then 100%.
In your case, specifying PAD_INDEX is not necessary, since by default your leave pages are how you want them. Full.
Best Answer
It's not clear to me how your first paragraph relates to the second, but I can answer the following bit:
When there is enough free space on the page for the record being inserted (whether in the middle or at the end), there will be no page split, as the page can simply be rewritten with the new data inserted.
If there is insufficient space, then there will be a page split, with approx half the data moved to a new page.
Note that if the insert is to the end of the very last page, and there's no space, then a new page will be created just for the new row and subsequent appends. That's what's known as a "good" page split, though it really isn't a split at all even though SQL server counts it as one.