To summarize David's great answer for folks perhaps looking for an answer more approachable to a wider audience: the answer is a thorough yes to the main question.
Hurricanes do have a cooling impact.
Hurricanes take warm air near sea level (and evaporated moisture, which is also a form of energy) and release it higher in the atmosphere where it is cooler (the secondary circulation David mentions)
All weather is really caused by imbalances, and the resulting weather processes naturally work to remove the imbalance... the basic idea behind thermodynamics boils down to the concept that the naturally "desired" state of things is evenness... in other words things seek to smooth out (for example, if you put hot and cold water together, their molecules collide and even out energy until the imbalance between them is gone... leaving only uniform warm water).
Is the heat converted into kinetic energy?
And yes indeed, a lot of the energy in hurricanes (and for that matter, many storm systems, and even vigorous thunderstorms) is transformed into kinetic energy (which ends up, at least gradually, being dissipated into the surroundings via friction/diffusion/etc). Though it's worth noting that in hurricanes the wind's kinetic energy (i.e. the strong wind speeds) are actually greatly responsible for acquiring more fuel... by evaporating moisture from the sea's surface (see this article for a deeper explanation of two central concepts to tropical cyclone energy). So even though the process gradually dissipates energy, it also winds up continually acquiring more such that those winds can maintain and potentially strengthen tropical cyclones...... up until the fuel source (warm moist water) gets removed.
Could the shape of the hurricanes reflect sunlight?
Yes, clouds, especially more opaque ones such as found in moist convection (i.e. storms, hurricanes, etc), prevent some of the sunlight and its associated energy from reaching the ground. This is primarily by reflection (particularly by white colored clouds), but also absorption and scattering. And these explain why it's typically at least a bit cooler on a cloudy day. (Or perhaps all the explanation really needed is to think of how you feel cooler when a cloud suddenly shades you from the sun on a hot afternoon!)
Do they create more lightning in an area that would normally have
lower levels of lightning for the same time frame in turn creating
As the NHC notes, tropical cyclones are actually not known as heavy lightning producers. This is because the air aloft in hurricanes is warmer than most locations, and lightning generally requires significant ice crystal formation.
As you can see in this fantastic satellite animation of Hurricane Maria from the new GOES Lightning Mapper from its first year in use in 2017, there still can be a fair bit at times. There are actually specific situations/locations that encourage increases in lightning activity, and I'd suggest they fit into these categories:
- A developing system (less warm air has been released aloft to hinder lightning)
- Outer edges of tropical cyclones (less warm air has been released aloft in such areas)
- The eyewall of hurricanes/typhoons (where there is very strong rising motion which can cause increased particle interaction and height)
- Areas where tropical cyclones interact with land (where land's rougher surface means more friction, which leads to more rapid lifting motion and thus more particle interaction)
But that said, overall I wouldn't think lightning activity in tropical cyclones is all that different from normal environments in such regions.
Additionally, ozone released from lightning is put into the troposphere, not the stratosphere. So it may not work the way you may think of the ozone layer doing. The ozone layer absorbs most of the higher energy (UV) sunlight. That heats up the ozone layer... so following that process, ozone from lightning would actually make more sense if it warmed the atmosphere.
However, remember that most of the higher energy UV sunlight is already removed from the sunlight (by the ozone layer) before it reaches the troposphere. So remaining temperature effects should be smaller in the troposphere. Plus, the amount of ozone created is very very small... the troposphere only has a very small fraction of the ozone that the stratosphere does, even with continued lightning. (In the stratosphere, ozone is made naturally from oxygen directly by the more energetic solar radiation). And it appears that more ozone is actually made in the troposphere by pollution than by lightning.
So overall, I'd rate the impact of lightning in tropical cyclones on temperatures to be likely insignificant.
Regardless, overall hurricanes certainly have a "cooling effect" on Earth. Well, truly they're moving the warmth to cooler locations, so it's really more of a redistribution/mixing effect.
But in the end, the fact that significant radiation is transferred into kinetic energy and the fact that formed clouds hinder incoming sunlight both certainly work towards cooling the area near hurricanes.
However, as noted, the wind actually winds up counteracting this by continuing to draw in new warm fuel, such that hurricanes can persist for long periods of time (see Hurricane/Typhoon John) without weakening due to such cooling factors.
Still, there is also one additional factor that ties back in to your kinetic energy subquestion, which at times can more quickly bring harmful impacts to a storm: oceanic overturning. A strong hurricane can have rapid enough winds to cause significant mixing in the ocean that taps into deeper, colder water further below the surface. Check out this video from the 2008 hurricane season, or this NASA article involving two hurricanes in 1998. Now indeed, hurricanes really only start to hinder themselves significantly if they are strong systems and remain nearly stalled for a few days. And the wake region of cooler sea surface temperatures often doesn't linger all too long. But that is indeed one additional way yet that hurricanes do
cool the Earth redistribute energy/reduce imbalance.