Third party units and/or missing units (official or not) might affect stability. We recommend having all official units (3.1 patch, Core Hedgehog, Arm Scarab, Arm Flea, Core Immolator, Arm FARK, Core Necro). Check the required units from the Replayer before loading a game.
You can replay games for friends over the internet: people can connect to your replay server like a normal game, and they don't need the demo themselves (which would defeat the point). However replay will not be as smooth. The host must also be on a sufficiently fast connection to accomodate all watchers at the desired replay speed. Lower it if necessary. +bps is always helpful to detect bandwidth problems ahead of time.
When speeding up replay, performance will suffer as more units appear and the action accelerates; severity depends on your computer system. Every speed change in itself may hamper performance as well, regardless of whether you put the speed back to 100%.
Restarting TA between recording games may improve stability. Restarting Windows is strongly recommended after replaying a game. Well, it always helps to restart Windows no matter what you're trying to do anyway :o)