Saving

Jurisprudence is a big field of Islamic knowledge which has its own experts who spend decades of years to be able to study the evidence from Quranic verses after going through all the fields of Quranic studies, and the authentic Hadeeths after going through the fields of Hadeeth and Rijaal which deals with assessing the authenticity of every narration. We also have the principles of Fiqh which is logic of jurisprudential thinking. 
'You need to go through the evidence confirmed by the great jurisprudence scholars or Maraaji' of Taqleed to see the reasons of any Islamic rule including your question.

'In brief, Islam does not permit RIBA (usury) between Muslims, but not every interest known today is usury. When non Muslims keep your money and put is their business and give you whatever they everyone else from their profit, it is permissible for you to accept it. This is according to the jurisprudence evidence which has details in jurisprudence books written by the Maraaje' of Taqleed. You are accepting what they usually give to every one who keeps his money with them as part of their business.

'Wassalam.

Yes. Khums is obligatory on us for any amount saved.

'Wassalam.

We Beed to look after selves as well as our family and children. We need good deeds for our future as much as we can. We also should for our family and children Whatever they need to have a respectable livelihood till they are able to earn for themselves. Obviously, we believe that Allah (SWT) will never leave them as He never left us.

Wassalam.

Yes it is allowed for every Muslim to invest his money after being sure that money will not be invested in unlawful business i.e. alcohol, interest, etc.

Khums will be applicable on the investment on the date of Khums.

Wassalam.