Hi and welcome and well...yuck!
I once had a kitten who used to wee in my shoes (and there was a joke on here the other week about 'Pissinboots' lol.) A friend's python used to regularly wet the bed - not his bed, his owner's bed, and if you've ever smelled python wee...
However. I'm not an expert and there will be more experienced people on here to advise, but just for starters I'd suggest that she obviously loves you and is determined to 'own' you - possibly the reason for her 'marking' your side of the bed.
On the other hand there is quite a houseful - dogs, kids, and she could feel quite threatened with all the different smells/noises. I agree socialisation has to be gradual here.
However, it's possible she's bored, in one room and especially at this time of year. I can understand your girlfriend's fear of her weeing in the children's beds, but could their bedroom doors be closed and the dogs put into one room/garden/walked while you let her explore, and play with her. The kids might enjoy helping with this, gently though, no screaming or chasing. (I know what kids are; I was once one myself.) Scrunched up bits of paper for her to chase, one of those tiny laser cat toy torches for her to run about after the beam.
Also - her tray. My youngest cat INSISTS on her tray being SPOTLESS. One wee in it - and she goes and makes a mess in the BATH!

Also the litter has to be fresh. New! I thought we'd got over this problem but in the past 2 days the bath has been visited ...animals eh? So although I'm sure you're fussy about a tray, especially in a bedroom, it's just a point. Also, if she hasn't got one, the hooded trays can make a cat feel more secure.
As I say, other people will have better ideas and more info, but do hope you get to the bottom of this...errm I'll rephrase that...do hope you resolve this problem, paws crossed and good luck!
