Can I create network with iPad & mini?

First, be sure the ATT wireless modem allows more than one device to connect at a time. A modem is not normally meant to act as a router and share the connection, although some modems with a WAP (wireless access point) built in can do so. Also check what they charge for the modem with wireless, as it may be cheaper to just get your own wireless router to plug into the existing modem.

As for security, 802.11g devices aren't going to be any less secure than 892.11n devices really, so long as you can use WPA. Both the mini and the iPad should support WPA fine. In a broader sense, nothing wireless is perfectly secure, since anyone that is near enough to receive signal can capture the data transmitted. It may take a lot to compromise the security of that data and the network at this point, but no form of security is eternally infallible.

When it comes to speed, your biggest constraint is going to be your connection to your ISP (internet service provider), in this case ATT DSL. Unless you are getting a fiber service, you probably aren't even close to the 54mbps speeds 802.11g can offer. The only improvement in data transfer you'll see by having 802.11n is between devices on your network. This means that your mini and your iPad can transfer things between each other faster, but neither can get data from the internet any faster than your ISP delivers.

It is not really that simple to upgrade the mini to 802.11n, and as noted may not really be worth it as it will likely not offer any gain in what you are doing.

Your house wiring to the telephone network only uses 2-4 conductors, so Cat5 or Cat6 wiring really won't make any real difference. Those are 8 conductor network cables, meant for wired networking.

An option you don't seem to have considered is instead of getting a wireless capable modem or router, you could simply share the internet connection from your mini via the Airport inside it. Since you are apparently wired from the modem to your mini, you can simply then use the mini's built in wifi to act as your WAP for the iPad.

Ok, I have a similar problem/question
I have my Mac Mini in my network via Ethernet. I also have a wireless network from the same router, but the signal is too weak for some areas of my house. So I have enabled Internet sharing on my Mac Mini.
Works great, good signal now in all rooms.
However, none of the devices connecting to it asked for a password - does that mean that anyone can get access through my Mac Mini now?
How can I password protect this? Do I have to?
My "normal" WLAN is password protected with WPA2

As the Mac mini is connected to that router via Ethernet, does it use the same password etc and my devices don't ask for it because the network name is irrelevant, as it is the same router for both?

Sorry, may be a dumb question, but I am not so much in technical details, I just want it to work conveniently and safe ...

Thanks in advance for any advice/help

Stefan