Hello, my name is Ben and it is my pleasure to assist you with your question today.
Did you perform any work in January?
When did you actually receive the notice to terminate your employment?
Only on the 24/12/2013 by e-mail
What was your contractual notice period during the probationary period?
I think 7 days , not sure
The dismissal will be valid. If the contract only required that the employer provides notice in writing, then sending an email is giving you notice in writing. As you were subject to a 7-day notice period, by giving you the notice on the 24th and having paid you up until the 31st, your employer has given you the notice you are legally due.
The issue then is the money they paid you for January. As your employment was already terminated and you did no work for them during January, this is not money you are legally entitled to and the employer has the right to ask for it back. If you refuse to return it voluntarily they can potentially sue you. There are limited circumstances when you could try and defend such a claim if you did not know you were overpaid and had already spent the money but assuming you were notified by receiving a payslip and the money then followed in your bank account you would have known about it so cannot use that excuse. As such, they are likely to succeed in a claim to recover that if it goes that far.
How comes that I still have no P 45 up to this day yet
they should have sent you one but it does not change the validity of the dismissal. It is bad practice on the employer's part but it is post-termination admin that does not mean you are not dismissed if it is not received
As far as I know I would pay emergency tax should I start a new employment. Who would compensate me for it?
this will be refunded to you by HMRC once your tax code issues are resolved
Please let me know if this has answered your original question or if you need me to clarify anything else for you in relation to this?