Connotation: "warm" is not the same as "hot". To illustrate:
cold == kalt
cool == kühl
mild == mild
warm == warm
hot == heiß
And in trying to develop this answer I learned this, which has marvelous opportunities for misunderstandings: schwül means muggy, sticky, oppressive, or sultry, primarily in a meteorological sense. Without the umlauted "u", schwul means homosexual, swishy, gay, or queer.
Pre-18th century, there was no schwül, and in Low German schwul had only the meteorological sense. So, keep that in mind next time you're reading ein deutsches Büch from before 1700 or so.
Yes it does, although I'm pretty sure the English "hot" is a lot more common than its German equivalent. Many younger people use "geil" instead (especially "geil aussehen" = "to look hot"), but if I had to choose between the two, I'd choose "heiß" over "geil" any day, it's much safer and much less controversial ("geil" was originally purely sexual -very sexual- but has since come to also mean "hot", "awesome" and pretty much anything positive in youth language. It's used to describe people, activities, things,... I've even heard it used to describe food. On the plus side, it will make you sound like a teenage native speaker immediately, on the downside, you might offend some people).
They do look similar, but no, there's no connection. Though in spoken German, "ich heiße" is often shortened to "ich heiß" :) (many people leave out the -e on verbs in the first person singular. Ich heiß, ich geh, ich mach, ich find... it's just so much more convenient, and saying the -e feels weird)
I am not sure if it refers to summers in general (as in "the SEASON summer is extremely hot") or if it is referring to a specially hot season ("THIS summer is extremely hot"). How would I know the difference? (since apparently I must always use the preposition when talking about dates and seasons). Danke!