两个expressions的意思

services rendered 提供的服务
It means services provided.

other interferences：
Austrian Cross for Services Rendered to Science and the Arts..
奥地利科学与艺术服务奉献十字奖章
eg：
We will pay all the specialists for their services rendered to us.
我方将支付所有专家为我方提供的服务费用。

family mythologies
Here mythologies means a body of stories about a person, institution, etc.
family mythologies means a body of stories about family.
or the similar expression is family legend.

You've rendered me a service.
你帮了我的忙。