You should install the Children one. My understanding is that GOG is based on European version of Fallout which didn't/doesn't have children. See here
http://www.nma-fallout.com/forum/viewtopic.php?t=59651&highlight= and
http://www.gog.com/en/forum/fallout_series/children/page1 if you care to do a bunch of reading, but yes you want the children patch.
AP Ammo patch is highly disputed. Some say it fixes the issue, others say it does nothing. Ultimately though both sides agree that the issue is with the code itself (in falloutw.exe) so changes to items will NOT fix it entirely. The only true fix to the AP Ammo issue would be a hex or assembly edit. For me personally, I go ahead and use the items, but don't assume it'll make AP ammo great -- you probably won't notice a difference.
Always always always use newest version of the hi-res patch unless you encounter issues specific to it (getting a black screen instead of the intro movie, that sort of thing). Each version he adds new stuff and fixes other stuff. Current version is 3.0.6 -- as PlanHex said it's up to personal preference, but if you're going to use the hi-res patch always use the newest version. Me personally, I never run at anything but 640x480 but I use it anyway
because it fixes the issues with the colors going all crazy if I put it into 16 or 32 bit mode, which is reason enough for me.
Same goes with sfall/ddraw - you don't technically need it, but being able to boost the game speed (Fo1 runs sluggish on some systems for no reason) and use of the mouse wheel is reason enough for everyone to use it.
I would personally highly recommend the TeamX 1.3.5 patch. Their NPC patch is a matter of personal taste -- without it you can't get your NPCs to change armor and having them change weapons is annoying, but it's not at all necessary to play the game. Whereas the 1.3.5 patch fixes a bunch of issues with the game, so you really have nothing to lose by installing it.