Enivob
Drop down a color node after the solver and choose random by attribute. Typically you can supply id as that attribute.

tamte
Create uvs on your geo before you plug it to vellum constraints or even copy to points, no need to do it in DOPs

tamte
either put @Time into Global Scale parameter (instead of Probability), which would do what Matt said, a Multiplication by @Time
so they will grow with different speeds

or

if you want them to start growing at different times set by your randomize, you can append Attribute Adjust FLoat after Attribute Randomize
- Attribute Name: zscale
- Attribute Class: Primitive
- Constant Value: @Time - 1
the (-1) is there to negate the maximum extrusion so that they start growing from 0
so if you increase Global Scale of Attrbute Randomize to 5 to give more randomization in time they start growing, adjust Constant Value to @Time - 5 etc.
- Enable Post Process: on
- Minimum: Enabled and set to 0
this will limit any negative scale so they will wait for their time to grow

or

you can also combine the approaches and make them grow at different times with different speeds

mestela
Multiply by @Time?