If you're sure you're going to use the Spark Core, then get your hands on that as soon as possible. Even if you're not going to use it for this project, it's really awesome, and definitely worthwhile!
The application for coding the firmware for the Core will most likely be the online IDE. Like the main page says, It makes use of the Wiring language, which is the same as Arduino uses. This makes it easy to find examples. It can also work with C/C++ if I'm not mistaken.
The Google spreadsheets part will work with their software, which looks a lot like JavaScript (haven't really looked into it yet). Depending on how difficult you want to make things on yourself, the examples provided in the topic I linked to, will suffice.
The Core itself will be able to work without an internet connection. You will however not be able to use its network functionalities. That means you won't be able to send information to the internet, nor will you be able to control it over the web. If you can bring a wireless router, that might be best, since then you'll be able to create your own network. Another possibility is to use a mobile hotspot. Most mobile phones nowadays are capable of sharing their internet connections by creating a hotspot. You can most likely use that to connect your Core to the internet.
I'm not sure what you mean by 'model', but the Core goes on the breadboard. Since it'll get it's measurements from there as well, it should work as long as the rest works.
Honestly, I would ditch the idea of the flow meter. It was just one of many possibilities, but @peekay123 's suggestion is much neater, and should be preferred.
That's up to you to code. The Core is as brilliant as it is dumb. It can do a lot of things, but only the things you tell it to do. If you don't tell it to measure things, it won't measure things. If you don't tell it to send data, it won't. Turning on when you want it to definitely belongs to the possibilities.
The Core can be powered over micro USB using pretty much any USB adapter that provides 5V, 300+mA, preferably 500+mA. The Core itself can not be charged, since it's doesn't contain a battery. Although it's possible to use an external battery pack (those cell phone back-up batteries), it might drain those rather fast, depending on the capacity. You can even make it run of normal batteries, but I'd suggest you stick with the USB powered micro USB cable.
@peekay123 mentioned this already, but I'll repeat it just to be sure. Coding stuff, and making projects like this can be, and often is, a lot of fun. When you've never done so before, and you're under time pressure it can become rather stressful. Is there anybody you could ask to help/assist you, preferably with some experience with things like these? If so, then please do ask for help, since it will save you a tremendous amount of problems.