Widgets did a pretty good job of answering your questions (thanks, Widgets!), but I am going to answer them again to make sure everything is clear.
Real-time inventory tracking, where the Order Manager keeps everything in synch among your various sales channels, is optional. There are very interesting arguments for and against using real-time inventory tracking, but the bottom line is that it depends on your business model, and obviously yours weighs against it. Note that it is also possible to track inventory for some items, but treat others as always in stock.
Credit card processing is very flexible. You can either do the full charge in the shopping cart, just get a pre-authorization in the cart and let the Order Manager capture the funds when you are ready to ship (that's probably the most popular option among our users), or do all credit card processing in the Order Manager.
You will probably use the Order Manager for shipping. The Order Manager can send tracking emails to your customers and/or send status and tracking data back to 3dCart.
Likewise, you will probably use the Order Manager to handle your drop shipping. It can create the drop ship POs and send them to your vendors. It can split orders among drop shippers and items that you ship yourself as necessary. And its PO system lets you reconcile your drop ship and stocking POs against your vendors' invoices and send the bills to QuickBooks for payment.
Inventory updates for changes that occur in the Order Manager (e.g. new stock received, POS or phone sales, etc.) are sent to your shopping cart(s) automatically, and usually within a minute or two. Updates due to Web sales happen in the Order Manager when you import the orders, which is done manually (just a button click) as often as you like.
The list of supported credit card processors can be found
here. No special hardware is required, and we just work with your existing gateway account(s).
Most modern barcode scanners will work with the Order Manager. The only requirements are that the scanner input appears to the computer as keyboard input, and you have to be able to program a prefix and suffix into the scanner.
We do not recommend using a wireless LAN with the Order Manager. In fact, we recommend at least 100BaseT and preferably a gigabit Ethernet LAN. I'm not sure about the LAN over AC adaptors - it would depend on how fast they are and what the error rates are like. If you are stuck with wireless, you could use Remote Desktop, GoToMyPC, Terminal Services or a similar system, which should work fine.
FYI - a little over 100 of our roughly 2,500 users are using 3dCart.
I am glad to answer any other questions you have here in the Forum. Also feel free to give us a call at 610-994-3699.