This is a vast topic and your question could be nominated for closure for being too broad, but I will make a few extended comments. It is not intended as full answer. People write whole books on this topic. For example, see the work of Judea Pearl, Sander Greenland and Miguel Hernán for starters.
Is it possible to use regression to check causality?
Certainly a regression model can be used to test a causal hypothesis, and this is done very frequently indeed. However, it is impossible, to prove causality with any statistical model. A researcher may hypothesize a particular causal path, and then use a regression model to test their hypothesis. The problem here is that the causal hypothesis, for example, "Does X cause Y ?", cannot be directly tested in the sense of getting an answer "yes" or "no". In frequentist statistics, for example, the test will usually provide an answer to the question: "if there is no association between X and Y, then the probability of obtaining these data, or data more extreme, is x%" which obviously is not the real question that the researcher has. This is a very common misunderstanding. On the other hand, if this probability is very low, then it is consistent with the researcher's causal hypothesis.
Directed Acyclic Graphs (DAGs), sometimes also known as causal diagrams are a very useful tool in causal inference, particularly in the context of regression analysis, and they can help to reduce a multitude of biases that may occur when trying to estimate a causal effect using regression. See this question and answers, for details:
How do DAGs help to reduce bias in causal inference?
Does this always hold "no causation without manipulation"? In other words, we can only talk about causation only after doing some sort of manipulation??
No. There is a large body of research on causal inference in observational data, where it is not possible to manipulate or perform an intervention.
It is true that a manipulation, or intervention, is strongly preferable to a purely observational study, however this is often not ethical. For example, we might be interested in the causal effect of some potentially harmful exposure on some health outcome, and it would be unethical to devise an experiment where some of the study group were exposed and others were not. Another reason why we can't conduct an experiment might be practicality. For example, we might be interested the effect of certain "trajectories" of childhood bmi on later life bmi. It would not be practical to devise an intervention to manipulate BMI in one arm of a study.
As mentioned above this is a vast area. Check this question and answers for further details about manipulations:
Statistics and causal inference?
