First thing I'll say is get your hands on a copy of Data Trails if you haven't yet. That book goes into awesome detail as to just what the Matrix, hosts, and such are and how they work.
So I'm gearing up to run a game and I was building hosts to confound my players. I attempted to logic my way through what a host might be like when a few questions cropped up that I could not answer.
1. Why do hosts have sleaze? Is a host smart enough to lie to a human? Heh. Maybe it could be used to upload spam or malware. But in practice for a corporate host, can I typically dump this stat?
Because hosts can run in silent mode. Also, any file that is hidden on the host would probably use it's sleaze rating to stay unnoticed.
2. Where is the matrix stored? By that, I mean does a search agent have to search the whole matrix? Are parts of itncached? If so, where? Capacity may be nigh infinite, but bandwidth certainly isn't.
Bandwidth in the wireless world is almost infinite. The two ways to signify to much traffic in an area is with noise, which also is used to represent lag from distance or signal blocking material, or being in a spam zone (which I think is in Data Trails). The information itself is stored everywhere, and nowhere. Every device in the world has large storage capacity and works on a distributed computing model. While your commlink is in your pocket not doing much, some of it's storage and processing ability is being used by hosts around you. Now that data is linked to that host so you can't just open up a file directory and see what's there.
3. Suppose I wanted my own host, but I run a modest criminal organization or am an anarchist and don't want my stuff perused by a hosting company, what physical form would the hardware take? Servers? Tiny servers?
You would register a host under fake credentials. I'd imagine few host making sites would be able to (legally) look at what's in someones host, especially since once a host is made you, not them, have ownership of the host with all the root access command that implies. Though if you get caught doing something illegal, I'm sure though it's not to hard to for local law enforcement get the authority to seize your files. But as stated before, there isn't such a thing as servers anymore, unless you choose to go with legacy tech, and then you're not part of the wireless matrix since the two run on different base code. Though a run to get data from an old legacy system would be a good/fun mission!
4. If my physical matrix hardware gets captured, can I build a self destruct? How common would this be?
You could probably set a host to auto-delete, even use a Nuke-From-Orbit program, but I would imagine this would be pretty rare and only used for some deep-dark stuff.
5. If a host is captured, can I set it to physically infrared beam its data to an offline backup once a day and to fry itself if captured? That would drive my players bonkers but is what I would do if I cared about my security.
Infrared?! That is so retro

But you can get offline storage and upload things to it. The details of how much this would cost and time are left pretty vague and up to the DM.
6. Archive rules sound sloppy and infallible. Why not keep everything not currently being worked on in archives? I intensely dislike this rule.
Probably because bringing something from or putting into the archive takes time and permissions (a full three marks) most employees won't have. Think of it as putting something in a safety deposit box in a bank vault. It's really secure, but a pain in the butt to get in and out of the vault if used everyday, especially when you have to get permission rubber-stamped by other really busy people.
7. Can a host be attacked/bricked directly? I guess that would be like a really aggressive dos attack.
Kinda, but not really. You can "attack" a host and get marks on it so you can get control of things, but you can't truly destroy a host in such a way. It's simply too......big, of a matrix presence for a decker to have that kind of affect on. That said, if you're willing to take the risk of a deep dive to the Foundation (Data Trails), you can destroy a host though there. Just fair warning, even against a small/low rating host you'll pretty much need to rank in the status of the Legendaries to pull of that righteous of a hack with your brain intact.
8. Can I pay to have my stuff hosted by an AAA mega behind their firewall? I guess this would subject it to perusal by said mega.
Maybe not the AAA's (though they probably have subsidiaries that do), but I would most definitely say that there are probably A and AA corps dedicated to secure data holding that have massive Firewall ratings and glacier thick IC. They've existed for the old Matrix, don't see why they would for the new.
9. Are there shadow/blind hosts that will host data for a price/cause? How safe is data being transfered overseas?
I'd say same as above, only rather then being a corp, it's a criminal syndicate who has ownership of the storage host. You don't really transfer data "overseas" like you do nowadays, or even with the old 'Trix. Moving from host to host is usually a fairly simple thing, though it could be intercepted while passing through whatever grid it's on between hosts. To be really secure, I'd suggest a courier for a high end firewall to hand deliver, as it were, the data.
10. Data on hosts can be hacked, but what if you just wanted to watch traffic coming to/from a host? Could you log it all encrypted, then break in and steal/exploit a control node to decrypt it all?
With the protocols GOD has in place, auto-surveillance devices don't last long. Data Trails has a program called Fly On The Wall that will let you hang around for longer without being noticed. You're generally limited on the time that you can keep an eye on things like that, though a cunning decker could still pull it off.
And a few more questions about the wireless world in general
11. Cyberlimbs can be used wirelessly anywhere. Are they actually accessing the matrix, or are they talking to your commlink? It seems like noise never affects smartlinks, limbs and other personal belongings. If it's talking to the matrix, is your limb accessing an external host? That would mean it needs its own account. Can you, with sufficient knowledge of a person, lock out all of their cyberware and device accounts without ever going near them? Can you jam that host's output at its source, depriving the opposition of wireless, but leaving yours intact locally? Brutal.
That's pretty much it. I know some of this might be abstracted within the rules, but it is very likely that one of my players is a security professional and these will be his questions.
While it connects to the matrix in theory, in practice all your wireless gear interacts for you/your commlink without worry about signal noise. They talk to each other directly rather then over the grid is I think the best way to describe it. I think jammers can still kill your wireless bonus for a time, but I'm not sure and will have to look it up.
Hope all that helps!