No online survival mode games for MW3

The lock icons indicate that when you complete that challenge, you'll unlock a new title or emblem. There's a "square" lock which indicates an emblem, and a "rectangular" lock which indicates a new title. It does not indicate a locked challenge. (I believe challenges you can't complete due to level or a missing unlock are grayed out instead)

You should be able to complete any challenge by the time you reach the 80th level of first prestige. Some of the pointstreak reward challenges require that those pointstreaks be unlocked in the relevant pointstreak menu. Note that if you're already at the max level, the only thing you can do is unlock the title/emblem, as gaining more XP means nothing at that point.

If you aren't making progress towards a challenge although you're sure you're doing it properly, this sounds like a bug. Also, you should only be able to level up to level 80 once per prestige. Technically, you have to get to level 81 in order to prestige again, but 4 times seems a bit much.

You may want to post your issue on the Technical Support forum of the Modern Warfare 3 forums. If there's an issue that requires the MW3 developers or community managers to get involved, that's the best place to post.

I don't know about installing from the disk, but my MW3 on steam saves seems to be inside the steam folder itself:

[path to steam]\steamapps\common\call of duty modern warfare 3\[profile name]\save