So would be possible to have a Hot (Tropical or Arid) North pole and a Cold/polar South pole with a certain planetary set up or geographic set up. I'm open to many ideas, as long as they make this setting possible. Edit: I forgot to mention, I want these area to be livable by humans, as well as the rest of the planet being livable. Would the rest of the planet change in order to do this or do the poles on need to change? THIS PLACE HAS TO BE LIVABLE BY HUMAN WITHOUT FUTURISTIC TECHNOLOGY
I have an idea here; it won't give you the exact effect you're looking for, but it can generate your desired poles at least seasonally. I'm assuming your planet has an Earth-like axial tilt, with seasons and so forth.
Giving your planet an eccentric orbit (oval-shaped, basically, instead of a nice tidy circle like we generally envision them being) and arranging it so that perihelion (the point in the orbit closest to the sun) turns up somewhere near the northern winter, will give you very different temperatures at your two poles. Your northern pole will end up with reasonably even temperatures year-round, because there will be a warmer planet overall during the northern winter to compensate for the reduced sunlight, and the planet will spread that heat to some degree. In the summer, the greater local sunlight is balanced out by a cooler planet overall. Tweak your temperature range properly, and you can reliably manage a warm pole. Your southern pole will be suitably frozen during the winter, but humans have lived in Siberia for thousands of years. In its summer, however, the sun is going to be pouring down the heat: how much in the way of an ice cap will survive, and how much that will compensate for the heat, is beyond my knowledge to reliably answer.
A side effect of doing this is that your seasons will not be of equal length. Your northern winter will be shorter than the northern summer (and the reverse in the southern hemisphere). You're also going to see planet-wide impacts on climate. Expect significantly more seasonal variation in the southern hemisphere (and less of it in the north). Also, the equator is always going to be warmer than the poles on average: if you want to make sure your north pole remains above freezing year-round, your equator is going to be scorching hot and probably very arid. You'll see some people living there, but most of your planet's population is probably going to be in the mid-latitudes.
Also, as a final warning: plants need sunlight to grow, not just heat. Agriculture near the poles, where sunlight will be basically nonexistent for several months a year, is going to be problematic no matter what the temperature is.
I will say no.
You specify that you are referring to the poles that are on the axis of rotation.
No matter where the poles are there will be no way to get a permanent summer at one pole and winter at the other. For this to happen, one of the poles would have to face the star all year round. You can think of the planet as acting as a giant gyroscope. Even if the planet's axis of rotation is in line with the sun at some point, half a year later the other pole will point at the sun.
Thus each pole will experience a summer and a winter over the course of a year.
Maybe there some other effect that can be exploited to make your scenario work. I can't think of one.
The effect you’re looking for is precession. Precession is when a spinning object’s axis of rotation varies it’s position over time, cycling round with a constant period. If your plannet precesses with the same period as your year then one pole will be in constant summer, the other in constant winter. They may still be colder than other places on the planet, but one pole will be significantly hotter.
Now: over geological timescales I have no idea if this arrangement can be stable, nor do I know if you can get a planet to precess fast enough to match an orbital period that puts it in a Goldilocks zone without some serious intervention, but at least a physical phenomenon exists that could explain your planet.