My girlfriend says that Rome fell because their plumbing was made of lead and it caused brain damage to the Roman citizens.  I said it was because the Roman Empire became overextended and they couldn't support the long supply lines to their armies.  Who is right?
Neither of you is entirely right or entirely wrong. The fall of the Roman Empire was a complex and multi-faceted event, and there is no single cause that can be said to be the reason for its decline.

Lead poisoning from the widespread use of lead plumbing and other lead-based products has been suggested as a factor in the decline of the Roman Empire, as lead exposure can lead to a range of health problems including brain damage. However, this is just one of many factors that likely contributed to the decline of the empire.

At the same time, the overextension of the Roman Empire, along with the associated challenges of maintaining long supply lines and supporting large armies, also likely played a role in its eventual downfall. Additionally, other factors such as economic instability, political corruption, and military invasions also likely contributed to the decline of the Roman Empire.

In conclusion, it is unlikely that any single factor was the sole cause of the fall of the Roman Empire. Instead, it was likely a combination of many different factors that contributed to its decline and eventual collapse.
How can one know, in any circumstance, which factors affect certain outcome? What's the most logical and sensible way to analyze situation, and determine all factors that caused the event and factor's weight in impacting that exact event?
In order to determine which factors affect a certain outcome and their respective weight in impacting that event, one can follow a structured approach that involves a combination of logical analysis, evidence gathering, and critical thinking.
By following a structured approach, you can logically and sensibly analyze a situation and determine the factors that contributed to an event and their relative weight in impacting that event.
Here is a suggested process:

1. Define the event or outcome: Clearly state the event or outcome you want to analyze. This will help you focus your analysis and ensure you are examining the relevant factors.
2. Gather information: Collect as much data and evidence as possible related to the event or outcome. This may involve historical research, reviewing case studies, analyzing data sets, conducting interviews, or gathering expert opinions.
3. Identify potential factors: Based on the gathered information, identify all the possible factors that may have influenced the event or outcome. Consider various perspectives and sources to ensure you have a comprehensive list.
4.. Categorize factors: Group the identified factors into categories, such as economic, social, political, environmental, or technological factors. This can help you better understand the relationships and interactions among the factors.
5. Assess the evidence: Analyze the available evidence to determine the relative importance of each factor in influencing the event or outcome. This may involve weighing the strength and quality of the evidence, as well as considering any potential biases or limitations in the data.
6. Determine causality: Establish the causal relationships between factors and the event or outcome. This may involve using logical reasoning, statistical analysis, or other methods to establish cause and effect relationships.
7. Evaluate the impact: Assess the relative impact or weight of each factor on the event or outcome. This may involve considering the magnitude, duration, or frequency of each factor's influence.
8. Check for confounding factors: Ensure that you have not overlooked any confounding factors or hidden variables that may be affecting the event or outcome. These factors can sometimes obscure the true relationships between variables.
9. Revise and refine: Continuously update and refine your analysis as new information becomes available or as your understanding of the event or outcome evolves.
10. Communicate findings: Clearly and effectively communicate your findings to relevant stakeholders, explaining the factors that influenced the event or outcome, their relative importance, and the evidence supporting your conclusions.