Is it necessary to change the name "Natalia" after its owner converts to Islam, knowing that the name means "Christmas Day" and does not involve worship of other than Allah?

The matter is as you have read in the fatwas: there is no obligation to change a non-Arabic name unless it contains something forbidden. If the name "Natalia" means the day of Christ's birth, then it should be changed. However, it is not obligatory unless it is exclusive to Christians.

Assuming it contains something forbidden, it might not be wise to rush in asking the new Muslim convert to change her name, fearing that embarrassing her could lead to her apostatizing. It is better to postpone this until faith has settled in her heart.

One must be cautious of being infatuated with her. It is better for women to call women to Islam and men to call men to Islam. Some righteous Muslim women can be assigned to teach her the rulings of the religion. If her Islam is good and marriage to her is possible, then do so. Otherwise, you should help her find a righteous husband.