Which battery chemistry to use in a project
I want to preface this with the following: I'm by no means an expert in the field of batteries and/or their chemistry. This is just what I know/have picked up over the years.

Depending what kind of autonomy you need, you need to decide what chemistry provides the best performance. 

If it is truly a very long-life product, you pretty much rule out nickel-cadmium (NiCd) and nickel-metal-hydride (NiMH). They have quite high self-discharge, and it is possible the self-discharge may deplete more of your capacity than your actual circuit does. In recent years, this has gotten a lot better with techologies like the Sanyo (now Panasonic) Eneloop type batteries. However, they are still outperformed by lithium-ion technology in most cases.

You have to also consider if you truly need rechargeable batteries. I'm not very familiar with the cost of modern (reliable) lithium-based technologies, but unless things have changed dramatically, it is very possible that it will be more expensive to use a rechargeable battery instead of replacing the battery every few months/years. 

Regarding discharge rates: This depends on the technology, but also the design. However, as lithium batteries tend to have lower ESR, they (generally) perform better in this regard.