Answers

Flashing is an essential component for any window installation.  This is part of the system which protects from water intrusion and structural deterioration.  There may be gaps between your framing or old frame and replacment window frame.  These are typically filled with insulation (usually foam) for energy conservation and to prevent drafts.

Simplified / general answer above.  As with most construction / improvement details, doing it right depends on products, conditions, and is more complex than a two or three line answer can fully describe.  Your best assurance of having it done right is to engage a responsible and experienced contractor.

Have them take that metal off that they installed "over" the existing step flashing. You'll need to check the step flashing cards to make sure they are all aligned and in the right places. Chances are they are not. If they are in the right places then it's your window leaking. Best I can do for you without running a hose on it. 

No you could run a hose on it...start along the roof line first and eliminate each area before spraying up towards the window. 

It appears that you are missing both step flashing (between the roof and the wall) and the head flashing (above the trim running parallel to the roof line). Here is the link to the Hardie Best Practices manual with specs on both flashings: https://www.jameshardiepros.com/getattachment/9a1017e1-853d-4574-b3e9-7afb1a5a472d/intro-tools-hz5-us-en.pdf. 

Each home is different, so it is suggested that you have it all inspected before making a decision as to what to do about it.

My best friend just had a similar issue, snow melt caused his to flood, it was not the cheapest fix, but a proper french drain dug around that section of the house was the soltion for him. 

Drylock is a product that is readily available at home depot brush corners roll walls can be applied to damp surfaces follow directions on can this should help

I would also check two things. Make sure all of the downspouts are running far enough away from the house. The ground is always moving. There could have been some settling that is now draining water back toward the house. Aldo, make sure all gutters are working proerly - no leaks.

If the ground seems to be diving down and back in to the home, bring in some black dirt to slope it back away from the house. This would obviously be the quickets and least expensive fix. Often times, it's the overlooked fix that would have solved the issue, rather than diverting water with a new wall system.

Good luck.

Chris

Having spent the last 24 years inspecting and waterproofing basements both responses so far are pretty good. The one thing I would add is you need to determine where the water is penetrating. If the water is entering your wall at or above the grade, then the second response from Lima Contracting may work. The only problem I foresee with that option is if it does not work then you will need to remove everything you just spent time and money doing. If the water is entering below grade, then you have to dig the wall out, including the chimney, and seal it properly. The waterproofing membrane that was originally applied to the wall has worn off and is no longer stopping the water from entering your foundation. This should be done prior to the winter freeze if at all possible. If you determine the water is above grade, I would strongly suggest excavating and sealing the wall prior to following Lima's advice. Chances are the below grade waterproofing has deteriorated and if you replace the concrete sidewalk without sealing the wall, you might have to tear it out in a year or two anyway. Lastly if the water is actually coming in through your floor or the joint between your wall and floor, you most likely cannot solve your problem without installing an interior sump pump. Hope the response was helpful. Bill

With this scenario.  The least expensive start is to on make sure all gutters are free from debris and working properly.  2.  Remove sidewalk and re-install new concrete all the way up to the house and brick pitched away from the house.  After concrete is cured in a week have a contractor come back and put masonary caulk between house/brick and concrete walway.

There is no short, easy, or inexpensive answer here.  In order to get a better handle on the situation and make appropriate suggestions, I would need to see more pictures. 

There is not much right with that set-up as it stands not though. 

Grading is bad to start with and grading and gutter routing are the two most frequent issues that I see when folks have basement water infiltration issues. 

The amount of mildew growth on the side indicates that this wall stays consistently wet. 

Post up some more pictures and I will give you some more pointed feedback and recommendations.