Another Eames LCW post sorry  

Answers
The wood is birch or ash (a little hard to tell which from the picture).
Yes, staining the chairs will lower their value.
Yes, you can re-glue the shock mounts. The factory glue is Scotchweld 1838; but PC-7 comes highly recommended in this forum, and is much easier to find.

Why would you want...
Why would you want everything in your house to be a uniform color? Isn't mixing everything together half the intention and half the fun?
I think you'd sorely regret staining them. When we first got our primavera and black Nelson bench we thought it would look horrible in our living room and were going to sell it but my god if it didn't bring everything together better.

This is the most ideal color
However, I will take your advice and leave the color as is.

Age?
What is the age of the chairs? They look fairly recent. The finish could be Light Cherry.
Walnut is not a stain, but rather a veneer (actual walnut wood).
P.S. The color of the chairs you posted is very similar to many Danish teak pieces.

matching wood is silly. i...
matching wood is silly. i have walnut cabinets in the kitchen with a blonde Eames dining set, walnut wall unit in living room with blonde kofod chair ... it all looks good...

New deadstock surplus
They were found in a local government auction and from my experience furniture is ordered, slightly used, and placed away when these agencies change decor. From my observations it seems that after many years the glue just breaks down overtime.

Do you have
A photo of the label(s)?

imo,
the thing with staining is a "close but no cigar" effect looks way worse than a definite contrast....think of wearing a faded denim jacket and trying to match exactly with a pair of jeans....

Cherry LCWs
From your photos, I think those LCWs are not stained, but rather have a cherry veneer. They also look like fairly recent production.
The probably stopped using them when the backs came detached. Are they marked with a Herman Miller label?

Just to clarify
The Cherry finish is a stained veneer, but I am guessing / assuming that the veneer itself is also cherry wood.... ?

They will be delivered
on Friday and I will snap a photograph of the labels. What I have come to find out with previous government auctions is that they tend not to have labels. I do not know the reasons as to why? However, they do tend to state the companies name etc..... Was this common practice back in the day? For instance I once posted some aluminum group chairs on DA that only had the date of production and the schools information.
I am curious with the business of labels.