I currently have a PCI 802.11n wireless network card (TP-LINK TL-WN951N 300M). Would a PCIe network card improve 1. download speed and/or 2. range (i.e. the bars that windows shows when talking about wireless connectivity in the bottom right corner of the screen)?
Regarding the wireless range, chipset and antenna play a major role, not the connection/bus technology. So a better card with an external antenna connector, with a better antenna, will have a longer range. PCI or PCIe, it's not important.
First figure out where your bottleneck is, I suspect it's your Internet connection (from your router to the Internet), and not a wireless link from your computer to the router.