When should you ult? Its situational. Ideally, you should analyze if the teamfight is something you or your team wants to commit to. Only if you're commiting to a teamfight should you ulti at the beginning of the fight. Sometimes, you don't even need your ult in a fight if you're already strong enough to deal with the enemy. Sometimes, you're not sure if your team wants to fight so you delay your ult until mid fight. In general, I will use ult at the beginning of the fight but only if I know my teammates are committing to the fight or I have the opportunity to instantly burst down an important target. This is something you will learn with more experience from playing vayne.
I usually run either
OR
if there is a major threat on the enemy team like
/
. It really is up to you if you choose to not take flash but flash has alot more uses whereas with exhaust, you're really looking to strengthen your 1v1 potential, even agianst a fed target.
As for best team comp to run her in, I personally don't see ideal comps. Shes going to scale really well and she wants to force 1v1s where she can dominate the enemy. A standard comp with hard CC is fine.
As for best lanes to gank, I generally focus top since its the easiest lane. In general, you always want to gank the lanes that are overextended where you have the dmg advantage (be very careful about ganking lanes where your ally is extremely behind). There are alot of factors to consider when deciding to gank a lane like "Can you kill the enemy or at least force them to burn summoner spells? Can you get an objective off of this gank?" Just keep in mind that shyvanna's pre-6 ganks aren't really strong and her really big powerspike comes from getting Sated Devourer. Once she has cutlass/botrk , her ability to chase down targets is much stronger. If you're going to gank as lvl 6 shyvanna with ult, be mindful of the hitbox on her ult. It has a very picky range that the enemy must be within in order to apply the knockback CC.
In general, master landing the 2nd shot of Double up on the enemy champions when you're going for poke. You'll make better use of ur mana pool and get better trades off. Also, try to cast Q between autoattacks because the animation can cancel your autoattack. With W, its better used for extended trades to maximize the effectiveness of the extra attack speed. The W can be used to give yourself a quick burst of movement speed if you're trying to escape from a fight or want to chase an enemy. The E is good now for slowing down the enemy which provides decent chasing/disengaging CC. A combo I'd recommend to increase the amount of burst dmg you deal with your ult is to cast: W -> aa -> Q -> aa -> ult ( will apply at least 3 stacks of impure shot). Otherwise, try to get impure shot stacks on the enemy before ulting.
With rumble, max Q>E>W. Take Q at level 1 and spam it to maintain at least 50 heat. Abuse the range on your Q by kiting the enemy and use both W and E to maintain your distance from the enemy. Regarding positioning of rumble ult, throw down slanted ults (adjacent to the general direction the enemy is running or in geometric terms: within 90 degrees of the line of direction) on the enemy in an open area. If they're running in very narrow areas/choke points, straight ults tend to work better. Also, you MUST take
. Rumble does insane AoE dmg once he has 1-2 completed items so a good teleport play can carry your team.