In elementary school (1940's) my teacher explained why the temperature varies during the different seasons, and I asked why it is generally colder a month or two after the solstice. Do you agree with my observation, and if so would you attribute this to slow changes in the temperature of the oceans, or to other effects? I also wonder why our geography lessons only included North America and Europe but that is another question that my teacher could not answer. Perhaps these two topics are not "research-related" but I would hope that your curiosity, which is essential for research, is quite broad.
We can start off with the simpler part: temperature of a region varies for a number of reasons. There are so many factors involved. Cloud cover is going to have a significant impact on the temperature of an area, as is its timing. If it's clear at night, temperature will drop, and it it clouds up before the sun has a chance to warm things up, the area temperature, all else being equal, will be lower. If things are reversed, you'll end up with insulation and solar warming.
Then you have to take into account weather fronts, patterns of air flow, temperature transfer from water and the ground. And as you suggested, there are heat sinks. And so you have a gradual change in longer term temperature trends. This answer is for the summer solstice, but the same argument applies to the winter solstice. http://curious.astro.cornell.edu/about-us/40-our-solar-system/the-earth/climate-and-weather/79-why-isn-t-the-summer-solstice-the-hottest-day-of-the-year-beginner
I don't know if you can tag people from outside a discussio, but I believe the article was written by @Jagadheep_Pandian
The ocean is a massive heat sink. How much it impacts temperature is going to depend on air and water currents, your proximity to the ocean, and so on. Also keep in mind that when the northern hemisphere is experiencing one solstice, the southern hemisphere is experiencing the other, so the distance from the equator is going to have an impact.
Just for fun, let's consider the following though (I'm sure it's been done many times): How much energy would have to be absorbed by the ocean, just for it to warm by one degree Celsius?
Ocean volume: 1.332 billion cubic kilometers if we accept https://www.livescience.com/6470-ocean-depth-volume-revealed.html
The specific heat of salt water is about 0.932 according to https://www.britannica.com/science/seawater/Thermal-properties
Let's say the density of salt water is 1030kg/m^3 https://hypertextbook.com/facts/2002/EdwardLaValley.shtml
So we have 1.372*10^21kg of salt water.
I'm lazy and am not converting, so I let Google do it for me: (0.932calories/1gram)*1.372*10^21kg = 5.35*10^24 joules
Let's see... in 2013 the total energy consumption of the world was about 5.67 * 10^20 joules according to Wikipedia. So about 1/10,000th the energy required to increase the Earth's oceans by an average of 1 degree Celsius. Oof.
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