Can Women in `Iiddah Go Out to Pray Tarawih and Go to Work?

We ask Allah to reward you in your calamity and to compensate you with something better. 

Can women in `iddah go out to pray tarawih?

The woman who is observing `iddah following the death of her husband should not go out at night except in cases of necessity. Your going out to pray Tarawih is not a necessity. Based on that, you should pray Tarawih in your house. 

Can women in `iddah go to work?

It is permissible for the woman who is observing `iddah following the death of her husband to go out during the day to work , but when night comes she has to stay at home. So there is nothing wrong with working in the grocery store, so long as that is during the day only.  

Ibn Qudamah (may Allah have mercy on him) said in al-Mughni (8/130): 

“The woman who is observing `iddah may go out for her needs during the day, whether she is divorced or has recently lost her husband, because of the report narrated by Jabir who said: My maternal aunt was thrice divorced, and she went out to harvest her palm trees. She was met by a man who rebuked her and told her not to do that. She mentioned that to the Prophet (blessings and peace of Allah be upon him) and he said: “Go out and harvest your palm trees, for perhaps you will give charity from it or do something good.” (Narrated by al-Nasai and Abu Dawud) 

Mujahid narrated: Some men were martyred on the day of Uhud and their wives came to the Messenger of Allah (blessings and peace of Allah be upon him) and said: O Messenger of Allah, we feel lonely at night; can we sleep in the house of one of us and then in the morning we will go back to our own houses? The Messenger of Allah (blessings and peace of Allah be upon him) said: “(Stay together and) talk in the house of one of you, then when you want to sleep, let each one of you go to her own house .” So she is not allowed to stay overnight anywhere but in her own house , and should not go out at night, except in case of necessity, because the night is the time when immorality is most likely to occur, unlike the day, which is the time when most people go about their needs and earn a living, and buy what they need.”

It says in Fatawa al-Lajnah al-Daimah (20/440): 

“The basic principle is that a woman should observe mourning in the house of her husband in which she was when he died, and she should not go out of it except in case of need or necessity, such as going to the hospital if she is sick, or buying what she needs from the marketplace such as bread and the like, if she does not have any one with her who can do that.”

Can you eat from the trees at the graves?

As for eating from the trees that are planted in the graveyard, there is nothing wrong with that, but you should note that the Prophet (blessings and peace of Allah be upon him) forbade women to visit the graves .

And Allah knows best.