OTA recording doesn't use a stream. The data goes straight from your TV antenna to a tuner in the AirTV black box, and thence to the attached disk drive. Your home network isn't even involved. So you can record one or two OTA shows without using any streams. The two-stream limit only applies to OTA (live or recorded) content you're watching at the time.
I'm less sure about Sling's cloud DVR (for recording Sling channels) since I never used it myself, but I think it doesn't use a stream either. It all happens on Sling's servers. Again, I think the 3-stream limit applies only to content (live or recorded) you're watching at the time.
I don't know if, if you have a good enough home network and five TV's, you can watch five things at once (two AirTV streams and three Sling streams). The Sling app might limit you to three simultaneous logins; if so, you could only watch from three TV's at once, two of which could be from the AirTV box.
The only downside is, on-demand content and policies vary from channel to channel. Some channels (e.g., TBS) are pretty good; others (e.g., MSNBC) aren't so good.
You can give Sling a 7-day trial before you buy the AirTV, to see whether the channels you watch have the content you want on demand, and whether using the cloud DVR counts against the 3-stream limit.