Fleas and flee eggs transfer to places on fabrics and your pets. If you have a pet with fleas, your pet will need to be treated for the fleas and you will need to take care of your living quarters as well before they will stop being a problem. Leaving will not fix fix the problem alone, unfortunately.
Yes, they will move with you. You need to either bug bomb the house your currently in, or when all your furniture and belongings are in a trailor to be moved bug bomb the trailor. Bugs dont stay put...they will travel with yoou.
Yes, they definitely can. You can try vacuum everything really really good and wash all bedding and clothes.
Yes they sure can