The biggest SD memory card a Nintendo Wii can support

Software-modding a Wii involves getting a copy of a game that contains a bug that can be exploited to run custom code. The first known hack involved a bug in Twilight Princess, for example, but the exploit doesn't work on Wiis with more recent firmware and other bugs had to be found. (I am not up on the current exploits, and being more specific would just make this answer out-of-date before long anyway.)

Bricking your Wii is unrecoverable without unusual technical skills. Bricking is also a risk you run any time you engage in soft-modding, so if you do decide to mod your Wii, do it only if you're willing to accept that you might destroy it.

Modding involves replacing parts of the operating system and firmware, so you can't update the firmware without losing the modding (and possibly having the update brick the machine in the process). However, the firmware has nothing to do with DLC like new songs and modding won't interfere with DLC. It's much more likely that a PAL game trying to access DLC servers in an NTSC region just won't connect properly, regardless of modding.

The default Wii browser is called the Internet Channel. It has limited support for JS and Flash, but I believe it provides some form of local storage.