From what I can tell, ults that only or mostly enhance other abilities will grant those abilities. For example, Shyvana's ult will transform and launch Sylas, and then he's Shyvana in dragon form until the effect runs out; Karma's ult will grant him one cast of one of Karma's empowered basic abilities; Elise's ult will turn Sylas into a spider with all of Elise's spider form abilities (including ult), until he transforms back (via reactivating ult); etc.
Nasus's ult reduces Siphoning Strike's CD by 50%, not necessarily any ability attached to Q. I would expect Sylas's Q to have its usual CD after stealing and using Nasus' ult, and also to not turn into Siphoning Strike (which wouldn't be much good without stacks anyway).
If Kalista is on the enemy team, Sylas starts the game with a Black Spear to designate an ult partner.
I would assume using Zed's ult successfully would grant the AD. I don't see why not. Keep in mind that Sylas is a mage with little concern for AD, though.
Udyr has an ability on the default R hotkey, but it's not actually coded as an ult. I don't know what would happen if Sylas tries to steal an ult from a champion who literally doesn't have one.
I assume stealing Kha's ult would grant whatever properties it has at the time, such as the evolution benefits. Same for Kass's stacks.
LB's ult is a boosted version of whatever ability she last used (or Sigil of Malice). Stealing her ult will most likely steal whatever spell she has ready there.
Anivia's ult ends when she's hard-CCed, moves out of range, or runs out of mana. I don't see any reason why any of that would change when used by Sylas.
Vayne's ult doesn't turn her invisible per se: it enhances her Tumble to grant invisibility when used. Given that Sylas doesn't have Tumble, and given that he doesn't have Night Hunter either (its strength is tripled while Vayne's ult is active), I would imagine that all he gets is the AD (which is apparently converted to AP/magic in some way so that he can benefit from AD/physical ults), but I wouldn't be surprised if there's an exception that gives him some part of those effects anyway, such as Night Hunter.