I'm not sure about the exact thresholds but it seems that the stars dictate how many affixes an item can have.

Legendary status adds to that further.

I noticed at least, that my 4star rare could have one more affix than my 3star rare.

So far I noticed it will only increase the base stats above all the affixes (that appear in white only). Whenever I do this I will get a slight increase in force shield/AR/health if the base is in white font.

If the item already has extra health/FS/AR as a magical affix (in blue) then I notice nothing changes, even if there is room for more stars to be filled in. And I have put in some items with only 3 affixes and it still did not come out with 4.

When I make a rare into a legendary then I will get more affixes, up to 6, if it had 4 to begin with.
I'm still testing this but before the patch if I upgrade a sub-5 star item to legendary then the base defense numbers did not change, so I figure for now if it's a good item I upgrade all the quality to 5* then increase rarity to legendary.

yeah its the same as adding quality to an item in path of exile with armour scraps or wetstones, it increases the base stat(s) the white version of that item has. physical damage on a weapon, all resist on a piece of heavy armour etc.