Is it possible to run the same set of independent variables with different outcome variables? I was wondering if it is possible to make any economic implications by regressing the same set of independent variables with different outcome variables. 
For instance, running [Industry, Years, Production Level] on [Revenue, Profits]. 
If it is, what are the appropriate regression models? Are there any concerned issues regarding the results?
Given the above variables, if both models are significant, what would be the implications? Does it imply that [Industry, Years, Production Level] can be used to explained both the change of [Revenue, Profits].
 A: Your question belongs to a well-studied statistical (or machine learning) problem, called multi-task learning, which

learns a problem together with other related problems at the same time, using a shared representation. ... It does this by learning tasks in parallel while using a shared representation; what is learned for each task can help other tasks be learned better.

The above states the rational of multi-task learning.
Depending on the type of your outcome variables, linear or logistic regression models are both applicable to multi-task learning. The design matrix (formed by your independent variables) will be used in separate regressions for each of your outcome variables, with the coefficients grouped across the outcomes. Therefore, this problem would be transformed into a grouped variable selection problem, which is among the hottest research topics in statistics in high-dimensional settings. Existing R packages (like grpreg) that conduct grouped variable selection (using methods like group lasso, group MCP, etc.) can be used for your purpose.
For your last part of question,

Given the above variables, if both models are significant, what would be the implications? Does it imply that [Industry, Years, Production Level] can be used to explained both the change of [Revenue, Profits].

The answer would be "Yes" based on your assumption. 
Note that under the grouped variable selection setting, it's possible to also formulate this problem as what's equivalent to bi-level variable selection. In that case, some coefficients of [Industry, Years, Production Level] will be allowed to be zero for different task, meaning different sets of variables explain the change of different task.
