Although ‘small’ works as a literal synonym for ‘wee’, a closer equivalent in English English real-life usage would be little, which quite often does not primarily refer to size at all.
Where ‘small’ is commonly used informatively for technical distinction (small, medium, large), ‘little’ can, by comparison, indicate some degree of sentimentality. (This is not a hard-and-fast distinction of usage: In the TV series Clangers, Small Clanger is the young son of the family, but affectionate big brother to his sister Tiny Clanger.)
Broadly speaking and by comparison, however, ‘little’ can often connote things like daintiness, delicacy or fondness. If I refer to my little idea, I mean something that I care about more than others might. If I talk about having written my little book (which might in fact be huge, physically), I mean that I am proud of it but I don’t assume that you will be impressed. The term ‘wee’ is often used similarly, although equally it can also simply mean literally ‘small’.
‘Have a wee seat’ could be smoothly rephrased in England as ‘have a little rest’. (‘Small’ would sound really odd in that sentence.) The wee-ness is something to do with being personally concerned for you. We might also say, ‘Have a quick break,’ but by comparison that would be relatively pragmatic.
‘Here’s your wee panini’ means something along the lines of ‘Here’s the panini that I am very pleased to be providing for you.’
I cannot swear to this, but I think that the same might apply in American English. A small house on the prairie would be, well, simply not as big as some others. A little house on the prairie is home.
I lived in Scotland for some years, and many times have heard Scots refer to ‘ma wee hoose’. Even if the house is actually quite big, it is ‘wee’ in the sense that the owner is quite fond of it, and feels protective towards it.
Your examples from Northern Ireland sound convincing (I have spent time there, too), and seem to reflect Scottish usage of ‘wee’ as well as (roughly) the English sense of ‘little’.