The raspberry pi was never really intended as a machine that you would use for everyday desktop computing tasks. It's more of a hobbyist machine for people who want to build do-it-yourself projects like a weather station, or a retro gaming console.
For comparison, I tried starting up Chromium on my pi-zero, which is a 5 years newer and has a 40% faster clock speed. It took 54 seconds to start up, i.e., about the same as what you observed. The fact that the faster clock speed has no effect suggests that the bottleneck is I/O, not computation. When I quit and restarted Chromium, it took 30 seconds to load, which supports this interpretation. Some or all of the data was cached in memory, so it didn't need to be read from the SD card again.
Typing the first character into the URL bar resulted in a 3-second delay for me, not 15 seconds. So here probably some other bottleneck is at work for you: maybe CPU or network I/O, depending on how the autocompletion feature is implemented. (Of course 3 seconds is still too long to be usable.)
If you really want to see if you can make your machine into a usable desktop, two possibilities seem to me like reasonable things to try:
Check the speed on your SD card. The speed is described as class 2, 4, 6, or 10. If it's not class 10, upgrading it would probably help with I/O-limited performance. (Mine is class 10.) However, the cost of the faster SD card might be comparable to the cost of a newer pi.
Look for software that is more specifically designed to be used on slow hardware with not very much memory. For example, there is a Wikipedia article Comparison of lightweight web browsers.
I am currently using a high speed cell phone charger adapter to power it (it's rated at 5V and 3A). Is this a bad idea, and would this be a reason for the slow performance?
This seems unlikely to me. The Pi 1B is supposed to require only 0.7 A, so this shouldn't be an issue for you, unless you add power-hungry peripherals powered through the pi. Per a comment from goldilocks, if your power supply couldn't supply enough power, your pi probably wouldn't slow down, it would probably just shut down.
Should I try installing a different distro?
Different distributions would probably not show a major difference in speed when running the same desktop applications. There are linux distros that are tailored for low-end hardware, but most of what they're doing is just changing the default window manager and installing a different set of desktop apps by default. The window manager you're using is already one specialized for this type of hardware, and you can install different applications without having to reinstall the OS.