There's no particular deadline to respond to a Rule 202 petition, except to respond before the hearing date set in the petition.

As far as getting more information on the Rule 202 petition, you'll need the cause number (which should be in the form 2007-XXXXX) assigned to the case by the district clerk's office. It doesn't look like you can use the links on the Harris County District Courts web site to search for the case by the litigants' names.

The Court is under no obligation to grant the Rule 202 motion. In this case, the movant must show that a failure to allow depositions prior to the filing of the actual lawsuit would cause a "failure or delay of justice." My personal experience is that these depositions would be used to perpetuate the testimony of someone who is seriously or mortally ill, who resides outside the U.S. and is available for deposition during a current trip, or whose testimony might be susceptible to "outside" pressure (e.g., an employer).

The pertinent portion of the rule is: "202.4 Order. If the court finds that allowing the petitioner to take the requested depositions may prevent a failure or delay of justice, it must order that the depositions may be taken."

I fail to see how this fishing expedition would prevent "a failure or delay of justice."

Re the Richmond/Westpark alignment lawsuit: You miss the mark on this one. Open government and the fifth amendment of the US Constitution and Art 1 Sec 17 of the Texas Const require that disclosures be made of plans to take or damage private property for the construction of a public facility. The lawsuit seeks disclosure.
The rail project should embed all the costs it imposes on private property owners as part of Metro's capital project costs. If it does not, it is shifting the cost of a public benefit to private owners. You would not like it if someone swipes your stuff; why should the government get to? Disclosures about exactly how much the private owners will be hurt and what sort of compensation is envisioned for them and when it will be pair will go a long way in reducing opposition to these projects and resulting lawsuits compelling such disclosures. A true property rights advocate would argue the stakeholders should be compensated to where they are indifferent as to whether the project is built. The extent to which Metro is not willing to commit ahead of time to this policy is a metric of the extent to which it intends to shift costs to the adjacent stakeholders.
My husband Bill Ware and I, definitely non-litigious people, filed a lawsuit against TxDOT and the FHWA because TxDOT would not reveal project information to us and they had missrepresented project impacts in an earlier lawsuit as well as to the COH. We have friends who are minority and small business owners along IH-10 (Katy Fwy) and they were promised to be compensated with moving expenses so the Katy could be built. I personally know of one who waited over 6 months to get TxDOT to pay his moving expenses. He was powerless to complain because TxDOT had conditioned his getting any moving expenses upon his execution of a release from bringing legal action.
While I am very supportive of CTC and other transportation organizations, this stuff has to be done within the parameters of the due process clause (notice and timely compensation).