Temperature of gas before CAT is from 350 to 900 degC.

The average temperatures are highly dependent on the application. Currently, most SI petrol engines with PFI fuel systems operate at, or near, a stoichiometric air/fuel ratio except during initial start-up or during high-load enrichment. Initial start-up exhaust temperatures are near ambient. Light-duty SI vehicles often use more than one catalyst substrate, often with a Pd-only close-coupled light-off catalyst mounted near or inside the exhaust manifold in countries with stringent pollution regulations to assist with cold-start catalyst light-off performance. Depending on the application, catalyst inlet temperature is often 650 C to 850 C at the inlet to the first catalyst. It is typically controlled to a 850 C to 900 C maximum temperature, often by fuel enrichment in engines with PFI fuel systems. The limiting factor for exhaust temperatures in the past had largely been exhaust catalyst durability (sintering of active metals and metal oxides and washcoat/substrate durability). Catalysts have improved to the point that the limiting factors nowadays are often the exhaust valve, exhuast port, exhaust manifold and, if the application is boosted, turbocharger turbine housing materials. Catalyst bed-temperatures differ from inlet and outlet temperatures due to thermal mass after startup and due to exothermic reactions during warmed-up operation. I have unpublished exhaust temperature data that is more recent but here is a link to some older published powertrain development data that includes exhaust and catalyst temperature data that may be of interest:

https://www.researchgate.net/publication/235343557_Demonstration_of_Tier_2_Emission_Levels_for_Heavy_Light-Duty_Trucks?ev=prf_pub

Although the paper is published by SAE, it is exempt from SAE's typical copyright since the work was funded in full by the U.S. Government. Note that the temperatures noted in the paper are often control set-points and although the paper is now 13 years old, the data is not very different from current production vehicles and their fuel system, spark timing and EGR calibration parameters. For example, spark-retard immediately following initial start-up is used to increase exhaust temperatures as an aid to achieving catalyst light-off. Enrichment was not encountered during urban driving (FTP cycle) but some enrichment was necessary during aggressive driving (US06 cycle in the example) to limit peak exhaust temperatures.

Older, carbureted petrol 4-stroke SI engines (still seen in industrial/nonroad applications) often have exhaust temperatures in the 600 C to 700 C range with fuel system calibration for power enrichment typically designed into the carburetor.

Article Demonstration of Tier 2 Emission Levels for Heavy Light-Duty Trucks

Would the chemical reaction inside a catalytic converter give rise to temperature?

Is it exothermic? 

Reactions over a 3-way catalyst at near stoichiometric air-fuel ratio are exothermic but gas residence time within the catalyst is so short and mass-flow rate through the catalyst is so high at anything above very light loads that temperature differences between the catalyst inlet and catalyst outlet are often less than 10-20C.  For diesel or lean-burn applications, there is excess oxygen in the exhaust and the catalytic reactions can be considerably more exothermic, although it also depends on both catalyst activity and reactants available for oxidation.