Hello,
1) She can certainly retain the property she purchased from compensation received after death of her husband. Her marriage with someone else is not disentitling her or providing any new right to relatives of husband.
2) She does not require to change her name under which she has the property registered. In case she chooses to sell or transfer the property at a later stage an affidavit will have to be attached with with relevant I'd proofs for clarification.
3) Name change is not required hence.
4) The relatives as indicated cannot make any claim of she married.