Firstly, the Terms of Employment Act (information) Act 1994 legally requires that an employee receives a contract within 2 months of starting employment. The Payment of Wages Act 1991 states An employer shall give or cause to be given to an employee a statement in writing specifying clearly the gross amount of the wages payable to the employee and the nature and amount of any deduction therefrom and the employer shall take such reasonable steps as are necessary to ensure that both the matter to which the statement relates and the statement are treated confidentially by the employer and his agents and by any other employees.
In terms of notice, as you are not 'contracted' to the company, they have no contractual notice period in place; so you could leave in one day. Notwithstanding this, notice for one year's employment is legally one week.
If they ask you to leave earlier then the week, and you agree, do so with confirmation of payment of notice. They are not obliged to pay for notice is you have agreed not to work it; again however there is no notice stipulation in the absence of a contract.
Should you wish to hava a contract, quote the Act I have mentioned above and confirm that you want one.
In terms of your tax query, I'm afraid I can't help other than you should have received a tax cert at the beginning of the year which will have stated your weekly/monthly cut off points etc. You should check this against the payslips you have received for accuracy. Again, I would insist on payslips quoting the Act above.