That Slayer can take things to TAR. Why not a person then as well? Dying in TAR might indeed make it impossible (or at least much more difficult), because you are already closer to outside of time, so to speak.

As for the other powers Slayer got from the DO (obviously through the TP), hard to tell. I wrote a theory once that Slayer might have a rudimentary shielding ability, which e.g. allows him to shield a channeler for a few seconds (SH shows a much stronger version of this power). The reason is that Slayer doesn't seem concerned about trying to kill Rand, after he failed in Far Madding, although he had to assume that Rand isn't in Far Madding anymore and thus will be able to channel.

We will see. It's a pity that this one Slayer scene was deleted from TSR. Maybe, if it had been included, it would be obvious to more people that Slayer killed Asmo..