I guess that you are in estate agency.
The technical answers to your question are:
1) The company would have to change its legal name and it then gets a new company number. However if it has a trading name which is different, there is nothing to stop the new company using it.
2) If you have been made redundant, then you must have been paid for your notice period plus statutory redundancy amounts (as a minimum). That, I'm afraid, is the limit of the old companies legal obligation to you.
3) If you have been offered a new contract with a new employer (legally this is the case), unless the contract states more notice, you would be on one week's notice until you have completed 2 years continuous employment.
It's pretty immoral, but probably not illegal.