What is the climb/descent rate that triggers the TCAS up/down arrow?
The Minimum Operational Performance Specifications (MOPS) for TCAS II (RTCA DO185B / EUROCAE ED143) specify the requirements for TCAS II. 

It states that the arrow must appear when the vertical speed of the intruder is 500 feet per minute or more:


  2.2.6.1.2.1.6 
  
  Intruder Vertical Speed Arrow
  
  A vertical arrow shall be placed to the immediate right of the traffic symbol if the vertical
  speed of the intruder (as determined by the TCAS tracker) is equal to
  or greater than 500 fpm, with the arrow pointing up for climbing
  traffic and down for descending traffic. The color of the arrow shall
  be the same as the traffic symbol.


Source: RTCA DO185B / EUROCAE ED14 Minimum Operational Performance Specifications for TCAS II, volume I.

Whilst the MOPS are not regulations themselves, the  aviation authorities put them into regulations such that TCAS II needs to be according to the MOPS. 

In the US, the FAA has published Advisory_Circular AC 20-151A which states:


  The FAA ACO approving the initial
   installation of the TCAS II equipment must verify that the TCAS II processor design does not differ from the criteria 
  specified in RTCA/DO-185A or 
  RTCA/DO-185B as applicable. 


This means it is not up to the manufacturer to decide when to show the arrow symbol. It is required by the MOPS, so it has to be designed that way.