The use of enethe’s portal key

As of version 1.12.5 of the Ingress Scanner app, we can answer these three questions as follows:

1. There is a limit of 2000 inventory items; this would limit your number of portal keys along with other items.
2. When you're at this limit, or if you upgraded while above this limit, your ability to hack is blocked, meaning presumably, that doesn't happen.
3. It is very clearly possible to hold duplicates of keys, they are shown in your inventory with the title addendum "(n)" where n is the number of copies you own. When you scroll through the keys to make a link, each copy is it's own item card shown grouped with its like portal keys.

It's further worth pointing out that now when you hack a portal it is no longer necessary to not be holding that portal's key in order to receive a key, or a copy of the key for your inventory. E.G. without managing your inventory in any special way, you should slowly find yourself acquiring 2 or more keys for the same portal.

This version's server side changes have also made it far less likely that you'll get a key at all; I had to hack a portal I live almost on top of 20 times over two days to get a single key for it. I have yet to get the keys for 4 portals around me that were added days ago after I submitted them.

You have to go there and hack the (destination) portal to obtain the key (yes, you need to hack "friendly" portals anyway to get supplies), or trade keys with team members by dropping/acquiring them.

Also, resonators "drop" keys which are used for links when one end is blown up that you can pick up.