Rationally I have to question whether giving money to charities for use in third world countries is a rational decision at all. While there are obvious benefits to, say, exterminating diseases such as polio, measles, ect. I have to question whether many of these charities are even worth investing in at all. We may claim we are looking at X many years of better life, but we have to consider:
1) Does that take into account the fact that these people are now more likely to have OTHER problems, seeing as they aren't dying from this one?
2) Are these numbers totally made up? (The answer appears to be yes)
3) Does this take into account that some lives are considerably less valuable than others?
If I am rescuing people in Africa, their median economic output is going to be far below what I would get for rescuing but a single person in the US, and moreover, the person in Africa is going to accrue additional expenses; their economic activity is unlikely to vastly outweigh the cost of sustaining them, while the person in the US is far more likely to generate a great deal of additional economic activity beyond what is necessary to sustain them.
While it may sound cynical, if there are fewer people in Africa, then that means there are fewer people who are living in poverty; if more people there starve to death today, there will be less people starving to death there in twenty years if we save them and allow them to reproduce. Moreover, if you save one person in the US and make them self sufficient, that is a huge boost. Additionally from a personal standpoint, helping out someone locally is far more likely to bring me benefits than helping out someone in Timbuktu, and, rationally, if I'm spending money and have the option of seeing benefits myself, isn't it more rational for me to spend the money in that way, because then I can actually appreciate the benefits of my personal generosity? Doesn't that also encourage me to be more generous in the future, and allow others to be more generous as well? Someone who can't even make ends meet is not going to be of the same generosity as someone who is making $60k/year.
rationally, if I'm spending money and have the option of seeing benefits myself, isn't it more rational for me to spend the money in that way, because then I can actually appreciate the benefits of my personal generosity?
It depends on what you're optimizing for - whether you value helping people, or seeing people helped (or if you value both, which one you're focusing on with the donation in question). Being rational determines how you go about getting the things you value, not what your values are.
That said, your other arguments seem to be pointing at ...
Why give globally? Why give money? Why health charities? Why single-issue organizations? At first glance these all seem like arbitrary choices: what if I would rather volunteer, or donate to local charities? Why does it matter? It comes down to two distributions: cost-effectiveness and income.
DALYs per $1000
This shows the cost-effectiveness of a large number of health interventions, with taller bars in cases where we can avert more death and suffering per dollar. The shape of this chart is important: while we can do a lot of good if we pick an intervention at random or support a 'horizontal' effort that works on everything, we can do 300 times better by picking one in the top 10%. This is why single-issue charities make sense: you can pick one that focuses on a top intervention.
(Don't let the small bars on the left fool you: nearly every intervention on that chart is worth doing [1], some are just far more valuable than others.)
This only considers health: what about other ways of helping people? Political advocacy, development, literacy, human rights, why not them? The big thing health has going for it is that we can measure impact, which lets us choose only the best options. In other fields where we can't measure we could end up anywhere on the impact curve.
Let's look at another distribution:
So some people have a lot more money that other people, we knew that, right? But have a look at the scale. Someone earning at the poverty line in the US is richer than 90% of people. This is why giving globally is so powerful: small amounts of your money can mean a huge amount to people who have so much less.
Neither of these distributions are intuitive: we don't feel that rich, and charities all seem kind of interchangeable. But understanding them can make the difference between trying to do good and really succeeding.
(I first saw these charts in a talk by Toby Ord of Giving What We Can (GWWC). The data for the first chart, DALYs per $1000, comes from the DCP2. This was a project that, among other things, compiled cost effectiveness estimates for a very wide range of health interventions. I made the chart from the csv version of the data from here, excluding the ~60 interventions (of 171) that didn't have estimates. The second chart is straight from GWWC's website, and you can read the details there by clicking on footnote 4.)
[1] The median intervention there is $207/DALY, which roughly means it can give someone an extra year of healthy life for $207. Which is an incredible deal, that I think most of us would jump at. And it's less than 5 pixels high.
I also posted this on my blog.