no. you can only see the coins held by a certain key.

you do not know all the keys a wallet holds.

and if they use a mixnet anonymiser then they can move funds between keys.

What's public is how many coins each address has. What's not public is who owns which addresses. If somebody wanted to be auditable, they'd basically have to send all of their coins to their publically-known address after each transaction they did, and then people could verify that the person who controlled that address controlled at least that many coins. You'd also need to solve the problem of determining who exactly controls the address. I can tell you an address, but really the only way that I can think of to prove to you that I own it is for me to perform a transaction that you securely tell me to do (tell me in a GPG message to transfer X coins to myself, for instance, where you specify X), and then for me to do it.

For instance, the EFF has posted an address of 1MCwBbhNGp5hRm5rC1Aims2YFRe2SXPYKt, which if you click on you can see the history of donations to that address. You can see that some spends from it have happened, so it's likely that some of those transactions have "change" that still belongs to the EFF, or maybe they've just split their funds across multiple wallets for security purposes. But you can total up their receipts via that address, so it's possible to audit that they've received at least X bitcoins. But they could have received bitcoins in another way (perhaps a large donor wants to give anonymously, so they created a new address just for that person to give them them), and there's really no telling how they spent that money. There's also no telling what procedures the EFF has in place for securing their wallet(s) and what people have permissions to spend their bitcoins. Pretty much all you can tell for certain is that the address that was posted on their web site has received at least X bitcoins, not who put that address up or what has been done with it. (And I'm not trying to pick on the EFF in any way; they're just the first high-profile address that came to mind, and it's conceivable that they'd want to advertise that they've received X bitcoins so far in donations in an advertising campaign or something, and I'm just saying that they could do that in an way that people could verify.)

You can't if they're generating a new address per person and a new address at regular intervals. You would only know the funds for one address that is shown to you, and assigned to you.

Unless they choose to use one address for their entire account.