The rain was heavy for 2 days only. It's more likely when you're here to rain 30-60 min in the afternoon, or not at all.

The Cu Chi tunnels are doable. Personally I tell my family it's not worth it, even though I love military history. And note if you're tall or, ahem, big boned, you won't be wanting to try and get into any tunnels anyway.

I usually recommend a Mekong tour if you're going to do something like that. I think it's more cultural and better, although still touristy.

If you arrive the night before, or get up really early, you can just head to Pham Ngu Lao or Ben Thanh and book a tour with one of the regular cheap operators. Or if you're looking for the private tours, you can often ask the tour operators, and they'll hook you up with friends. This is not ideal, but might be a little better than just asking a guy on the street. I've only used the group tours, so not much help on that.

If you want to stay in town, the IC is fairly central and you can walk to the cathedral, the several museums, the palace, and any markets you want to visit. Also it's popular to visit many food stalls or small restaurants. Eat some pho, spring rolls, bun bo hue, etc. It seems to be popular now too that people take a food tour at night, riding on the back of some vespas as a group.