Is this true that the requirement to worship and to sacrifice to the Roman gods was put in by the Caesar only after Christianity emerged? I just encountered a statement that this requirement was put in by the Caesar as a response to Christians' being too aggressive (like they were attacking other temples and harassed the worshipers in them). Is this true? If yes, can you, please, provide some source.
Yes. It was Decius. But I cannot say if it was because Christians were being too aggressive. It could have been an attempt to gain public acceptance.
But the sacrifice test had been in use since long before Decius. For more details see this document by de Ste. Croix.
It must be kept in mind that in ancient Rome sacrifices were quite common for both personal and imperial purposes. Sacrifice contracts "negotiated" with the Gods were fairly elaborate and would run into pages (or rather scrolls/tablets). Some of these contracts required subjects/citizens to sacrifice regularly for the well being of the army (and later the Princeps --or Caesar).
However, there is a link between sacrifices and prosecution of Christians. See Pliny's letter to Trajan for an example. As Pliny's letter indicates, prosecution does not seem to have been driven by the state but by the people --someone had had to be accused before being tried. In practise however one cannot rule out accusations coming from powerful state officials who wanted public approval.
The letter also gives us some insight into the trial. The accused was asked to denounce Christ and confirm Roman religion by offering incense and food to the Emperor.
As de Ste. Croix outlines --such trials frequently involved sacrificing to the gods (and not necessarily in the emperor's name). But it was not warranted by law and seems to have had been a legal argument to strengthen the accusation.
Decius however made sacrifice mandatory with his edict of 250 A.D.