During winter operations and de-icing are the engines (jet engines) usually in operation? If not, what is the reason for that? Also I was wondering if the engine nacelles require de-icing since I have never seen them being spryed in any video footage.
Generally, it's not required to shut the engines down.
Sometimes you're deiced at the gate or just off the gate, and in those cases you've not started the engines yet. Or if it's necessary to remove ice on the bottom of the wings or from the bottom/front of the (extended) flaps, those would be cases when you'd shut the engines down for deicing. But most commonly, you taxi to a de-ice pad (parking area), leave the engines at idle with the bleed valves closed (to avoid getting deice fluid into the air conditioning system - even a trace of it creates kind of a stink), and deice with engines running.
The engine cowl is deiced NOT with fluid but thermally, tapping bleed air (upstream of the bleed shutoff valves, so it's always available) to heat them. Thus the fluid is not directed at the engines, but is intentionally kept away from them.
Most jet, turboprop and reciprocating engines can be running when deicing. There may be exceptions but I have not encountered any. Deice fluid into the engine is wasted fluid, and is avoided.
There are four types of commonly used deicing fluids and in general they are toxic. Ethylene glycol is the most common, and is commonly also used in automotive anti-freeze solutions, and is toxic. For this reason, inspiration of deice fluid is avoided, which means shuttering cabin fresh air sources. That is also why deicing is done on ramps to contain (and recover) the deice fluid, minimizing environmental impact. The principle toxicity involves dehydration of animals as their with degradation of kidney function and the central nervous system. Other alcohols in addition to ethylene glycol are also used, as well as corosion inhibitors, surfactants and dyes.
Engine inlets are typically heated with bleed air (some are using electric) to deice, and that not commonly done concurrently with chemical deicing. Some turboprops call for alternate induction air, using inertial/centrifugal separators to reduce the ingestion of denser particles, pieces of ice, etc.
Aircraft engines typically operate at cruise power and climb power through precipitation and while not common, sometimes turbine engines are exposed to chunks of ice being ingested. Engines can handle a certain degree of that. A small amount of warm, sprayed deicing fluid will not significantly harm most turbine engines. In fact some turbines tend to have icing occur on the initial compressor fans, and that ice is shed into the engine, usually without adverse consequence.
In summary, casual ingestion of deice fluid by aircraft engines is not harmful, and engines may be left running during deice operations. Engine inlet deicing is managed normally through application of bleed air. Some turboprops will use inertial separators to reduce risks from inlet icing and icing debris.