To solve this question, we need to analyze the given statements regarding India's response to climate change and determine the correct relationships among them based on the provided options.

1. Statement I: "India's Long-Term Low Emission Development Strategy (LT-LEDS) is a crucial tool for achieving net-zero emissions by 2070."
   • This statement highlights India's strategic approach to achieving long-term sustainability by setting a net-zero target for 2070. The LT-LEDS serves as a guiding framework for transitioning to low-carbon pathways over a prolonged period.
2. Statement II: "India's 4th Biennial Update Report (BUR-4) submitted in December, 2024 recorded around 8% decrease in Greenhouse gas emissions in 2020 over 2019."
   • This statement refers to a factual report indicating a reduction in greenhouse gas emissions in the year 2020 compared to 2019. This empirical data could signify progress toward emission reduction goals.
3. Statement III: "Climate-resilient development necessarily depends on quick and short-term achievement of emission reduction targets."
   • This statement implies the importance of immediate actions to achieve climate resilience, which seems to contrast with long-term strategies.

Now, let's evaluate the relationships among the statements as per the given options:

• Option 1 (Statement I is empirically supported by statement II):
  • Statement II provides empirical evidence—an 8% reduction in emissions—which can be seen as an indicator supporting the feasibility of the long-term strategy mentioned in Statement I. Thus, option 1 is correct.
• Option 2 (Statement III contradicts the approach implicit in statement I):
  • Statement III emphasizes short-term actions, while Statement I focuses on a long-term strategy. These two approaches appear contradictory where one focuses on immediate action and the other on gradual development. Thus, option 2 is correct.
• Option 3 (Statement I and statement III together establish the premise of long-term sustainability):
  • This option is incorrect because Statement I and Statement III have conflicting approaches regarding sustainability goals—one long-term and the other short-term.

Hence, the correct answer is: 1 and 2.