Short answer is yes, Android sometimes does lower the download rate when the screen turns off. According to this Android bug report (starred by 90 users),
It seems that, even with "Wifi Sleep Policy" set to "Never", the wifi on some Android devices will enter Power Save mode (a.k.a. PSP mode) when the screen turns off.
PSP mode apparently causes the wifi device to sleep, but wake up periodically and ask its access point to deliver packets that were addressed to to the device while it was asleep. This necessarily causes additional, "spiky" network latency. This additional latency means that some network-oriented applications experience problems when the screen turns off.
This more or less correlates with the behavior you're describing. This issue affects only some phones, not all, and only some versions of Android. I suggest you visit the bug report above, there is a long discussion of this issue.
How to solve this issue
There is a workaround for Android 2.3 at least: You need to acquire WIFI_MODE_HIGH_PERF lock:
In this Wi-Fi lock mode, Wi-Fi will be kept active as in mode WIFI_MODE_FULL but it operates at high performance with minimum packet loss and low packet latency even when the device screen is off. This mode will consume more power and hence should be used only when there is a need for such an active connection.
CPU clock scaling in Android
Android is based on Linux kernel, and uses CPUFreq Governors to adjust CPU frequency to conserve the battery. Exact policies and frequencies vary from vendor to vendor and are subject to change. There are also utilities that let you adjust these policies, but they all require rooted phone. Usually, Android phones lower CPU frequency and use "powersave" governor when the screen is turned off.
However, I don't think network traffic on a phone is CPU-bound. That is, even when in power-safe mode, network throughput is a bottleneck, not CPU speed.