Setup the document as Facing Pages, if it is not already.
For the chapter name, make sure each chapter heading uses a dedicated paragraph style for example, Chapter Title.
Then go to:
Type > Text Variables > Define
Create a new variable based on Running Header (Paragraph Style), and choose the paragraph style used for your chapter heading.
Then go to the parent page where your footer is set up, click into or create footer text frame, and insert the variable using:
Type > Text Variables > Insert Variable
That will pull the chapter name automatically from the styled heading on the page/spread.
Then style the Variable to your liking, font, size, colour etc.
For the design side, it is common to mirror the layout on facing pages page numbers toward the outside edge, and the chapter title or running footer toward the inside or opposite side. For example, on left-hand pages the page number might sit on the left, and on right-hand pages it might sit on the right. You can control this on the left and right parent pages.
There is no single rule, though. The main thing is consistency and making sure the page number is easy to find.
If you need further directions let us know.
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