You have a Bag of Holding, which you would like to store inside a small wooden chest. Can you fill the bag with various items, then fold it as if it were empty and put it into the chest? Or does it appear full when there's stuff inside it? How much stuff can it hold before it starts to look full?
The Rules Don't Say.
I never like giving one of these answers, but the simple answer is: it doesn't say one way or the other, anywhere.
D&D 1st Edition is where we first saw the Bag of Holding introduced, the description given is...
As with other magic bags, this one appears to be a common cloth sack about 2' x 4' size. The bag of holding opens into a non-dimensional space and its inside is larger than its outside dimensions. Regardless of what is put into this item, the bag of holding always weighs a fixed amount. This weight, the bag's weight limit in contents, and its volume content are dependent upon its quality as shown below.
If overloaded, or sharp objects are placed within so as to pierce it, the bag will rupture and be ruined, and the contents will be lost forever in the vortices of nilspace.
AD&D is much the same, but added the rules of what happens if the bag is pierced from the outside, and what happens if you mix bags of holding and portable holes.
D&D 3E added rules for the sort of Actions required to retrieve stuff from the bag, what happens if you turn it inside out, and what happens if you stuff a living creature into the bag.
4E Nerfed the daylights out of the Bag of Holding, but added no new rules or flavor.
5E rolled things back to closer to 3E rules, but without multiple 'qualities' or 'sizes' of Bags of Holding.
And that brings us up to date. The fluff text on a Bag of Holding has not changed since D&D 1st Edition. At no point has the interior of a Bag of Holding, or how it appears when being filled or emptied been laid out. Even branching out to Pathfinder, they simply lifted the description straight from 3E and left it alone.
What I would do...
My personal ruling that I use in my games, which has never really caused trouble...is that the bag appears roughly as full as it actually is. If 50% of its capacity is full, the bag appears 50% full. If it's empty, it looks empty. This has never really come up in terms of putting the Bag inside of something else...but has always been an easy way for characters in-game to tell how full the Bag is at a glance.
(For extra fluff, I've also always ruled that the inside of a bag of holding looks....like the inside of a bag would. Just really big.)
There are no rules as written, but here's my take for what it is.
A Bag of Holding always contains the same thing: a piece of another plane. It's like a bag with an inflated balloon in it. You're not actually putting anything in the bag, you're putting items in the plane. As such, the bag should always look like it's full because, as far as the bag is concerned, it is always full.
You can also think of the opening of the bag as a portal to another plane. Nothing goes into the bag, it goes into the portal. The bag contains a portal to that plane and nothing else.
The Bag Always Looks the Same (maybe half-full)
The rules say "the bag weighs 15 pounds, regardless of its contents", which seems to imply that the look and feel of the bag does not change as more items are put into it (i.e. it would look/feel the same when empty as when full). This seems consistent with the canon of it being a portal to a demiplane.
Now: the bag's illustration shows it to be about the same size/shape as a laptop/messenger bag (which don't weigh 15 lbs. when empty). To me that seems like it probably feels like it's always holding something. As to why, you'd have to come up with your own explanation.
The explanation I've used in my own game is: the bag has stuff (magical components/tchotchkes) sewn into the interior as part of its creation. However, users never see these components since they only see the portal when opening the bag (unless they open the bag in an anti-magic field).
For folding purpose, I'd probably treat it similarly to a normal messenger bag or large purse filled with about 10 lbs. of fist-sized items. So: probably fold-able once.