Problem: Regression with Fixed Effects (multiple dummies variables)

Dear all,
I want to run the following regressions with Fixed Effect for time (year) and firm identifiers (cusip).

SALARY the dependent variable
FIRMSIZE the independent variable
YearFE dummy variable (for categorical variable year)
FirmFE dummy variable (for categorical variable cusip)

The models are:

1] SALARY=α+β_a * FIRMSIZE + YearFE + ε
Based on:
xi: areg dependent_variable independent_variables i.year, absorb(firm_identifier)
I write:

Hopefully is correct (?)

2] SALARY=α+β_b *FIRMSIZE + YearFE + FirmFE + ε
Based on:
xi: reg dependent_variable independent_variables i.year i.firm_identifier
I write:

Q1] the calculated coefficients β_a and β_b are equal, is that correct? Is there an error with the syntaxis?

I am adding a new dummy variable:
CRISIS. CRISIS is a dummy variable that takes value 1 if the year is bigger or equal to 2008, 0 otherwise. So:
Then I want to run the following regression:

3] SALARY=α + β _c FIRMSIZE *CRISIS +γ_c *FIRMSIZE + YearFE + ε

Q2] Sorry but I don’t know how to write it… Any suggestions?

Thank you in advance for your help!!

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I am using STATA MP 13.1, in a Windows 10 PC.