No weave or channeling REQUIRES any gestures at all, but the Aes Sedai consider it an aid to learning, because they themselves were taught that way. Lews Therin required hand movements as well, so it's not just a third age thing (remember "I CAN'T SEE MY HANDS". Wise ones generally saw the uselessness of waving your arms around, for more than one reason - it slows your channeling (Aviendha's fireballs flew from her at an amazing rate when battling the seanchan attack on the Kin's farm), and you may not be able to move your hands (knocked out by a rock, then tied up, or losing a hand, like Rand). Problem is, once you learn a weave a certain way (or even randomly come across it) it's damn near impossible to learn it any other way (Aviendha has masses of trouble opening a gateway around ACoS or TPoD, because she stumbled across it in TFoH and forgot how she did it to begin with). Gestures are not required to DO the weave, unless you learned to do the weave by doing the gestures.

As far as the circle for Mat, they were breaking a "link" between him and the dagger first, and healing him second. I think it was more of a 'who's handy and trustworthy' kind of thing. Moreover, I don't think old healing would be any more effective in a circle, unless everyone was fairly weak in healing. Remember, old healing can kill the healed if they're so far gone, and more than one Aes Sedai in the series has hesitated because she alone could kill someone from healing them too rough.