Yes, she has the right to ask you for a receipt, and yes you should put you NIE on it.
This answer on the subject of contracts from regular legal-contributor Raymundo Larrain:
Short term let has no minimum number of months; that is something people made up.
Long term contracts: are when the house is needed permanently to live in (family, to work, kids go to school etc.). By law tenants can stay in the property three years (paying).
Short term contract: anything that does not fit in the above. i.e. summer let, holiday rental home (days, weeks, couple of months).
Eleven month contracts do NOT exist.