Is there any downside in broadcasting multiple wireless networks from the same router?
If you mean from the same radio on an AP/router, then absolutely. Each additional SSID requires a certain amount of overhead, at the very least in the form of beacon frames and other management traffic. This reduces the efficiency of the wireless network and reduces performance with the effects increasing for each additional SSID.
Most of the references citing statistics for multiple SSIDs of which I am aware of are older documentation. The numbers may not be as exact due to changes in 802.11, but the principles still apply. Data rates have increased, but so has the size of the typical beacon frame. Anyhow, here are references from Arubu's Airheads Community and Revolution Wi-Fi that provide statistics showing the impact of multiple SSIDs.
Is there an upper limit?
Different hardware/software may have limits that are imposed, but there is no limit as defined by the standards. However there are maximum recommended limits and the general rule of thumb is to use the fewest SSIDs as possible.
Cisco's best practices document recommends 1-3 SSIDs:
It is recommended to have one to three SSIDs for an enterprise, and one SSID for high-density designs.
Aruba's best practices give an even lower target at 2:
Use as few SSIDs as possible. Generally, one WPA2 Enterprise SSID and one Open SSID are more than enough.
Can different networks broadcast by the same router have different channels/frequencies?
Only if the SSIDs are using two different radios. If the different SSIDs are using the same radio, then no, they must use the same channel/frequency.
Note: Portions of this post are copied from my own response to a similar question on ServerFault.