You need tough love man.. get rid of her asap!! She is no good for you... you need to leave her and give yourself time to get over her at least. You sound exactly like me when I split with my ex after finding out on her facebook she was planning on meeting another guy. You had every right to check because of your gut instincts and history with the girl and that other dude.. it's probably the best thing you could have done. (though not recommended all the time lol)
The bike and money etc, go over unexpected and give the money and bike back to her mum when you can. Don't let off steam to the mum, just say you can't stay, things havn't worked out between you and your gf, so you need to give the stuff back and go do something, whatever that may be.. say thanks to her mum and go on your way. If the girl is there and asks to talk or anything, say you have plans and can't stay, then just walk away!
Now, this girl.. breaking all contact is what I would do. She has fuc*ed with your head, I can tell just from reading this cus you can't seem to put your point across properly and doubt yourself a lot. I'm not having a dig mate, just saying it as I see it, and it's bugs me to see this happening to you. Just get her out of your life and start moving on. She's treating you like sh1t man, you're being totally AFC and accepting her sh1t. Once she see's that you have no problem leaving her in your dust she'll feel it for sure. But whatever you do, DON'T give in to her begging you back or anything if she does. Text her saying she had her chances, it's over, and she should move on (telling her to she needs to move on puts you on the right frame of being the person who isn't chasing). Don't pick up her calls or reply to texts/facebook messages etc, and focus on yourself.
I'm sure you're hurting from this as it sounds like you were really into this girl, but yes you are totally justified in dumping her.. grab yourself by the balls, be a man, tell her it's over, and get out to have fun for YOU! I can tell she has messed with your head, and it's a bummer as I've been there (as have many, many other guys).. don't dwell, learn from this and make yourself a better man because of it!
I know this is all bit stern sounding, but I'm being straight cus I don't want you to let this affect you in a negative way. It was after a very simliar situation I got into pick-up... get into it too, and it will help you make sure this stuff is much less likely to happen, and make you better at dealing with it! 