Many of the major clans in the Sengoku period survived the transition to the Edo period. Of the list you named, Azai, Chosokabe, Toyotomi, Akechi died out or were wiped out. But the rest survived as minor lords in the Edo period.

After the Edo period, these lords became members of a new aristocracy called the Kazoku. Kazoku were composed of three groups. (A) former court aristocrats in Kyoto, (B) former daimyo lords, and (C) individuals who performed meritorious service.

After WW2, the Kazoku system was disbanded. Today former Kazoku families gather at the Kasumi kaikan. Many of the descendants are active in politics and business, but a few are in show business or are athletes and academics. Most people in Japan don't really care or know about the former Kazoku.

Hosokawa Morihiro (Prime Minister 1993) is a direct descendant of Hosokawa Tadaoki (his wife was Akechi Mitsuhide's daughter).

Oda Nobunari (figure skater) is supposedly a descendant of Oda Nobunaga.

Tokugawa Tsunenari (World Wildlife Fund Japan division director, former Vice President of Nippon Yusen, a major shipping company) is the current head of the Tokugawa clan.

Date Yasumune is the current head of the Date clan (of Date Masamune) and is a historian and director of the Zuihoden Museum.

Kikkawa Koji is a singer and is supposedly a descendant of the 2nd son of Mori Motonari (Choshu domain).

Kuroda Nagahisa (1916-2009) and Kuroda Nagamichi (1889-1978) both son and father were orthnithologists and were the heads of the Kuroda clan. The current head of the Kuroda clan is Nagahisa's son. The story of Kuroda Kanbei (aka Kuroda Josui) is currently on NHK as a year long historical drama.

Below is in Japanese but is a partial list of famous people whose families were Kazoku.

http://matome.naver.jp/odai/2137414493755415701

Let me end with the story of Shimadzu corporation. The founder and inventor, Shimadzu Genzo (1839-1894), has the same last name as the rulers of Satsuma domain, Shimadzu clan, and the two share the same crest. Today, many people think Shimadzu corporation is related to the Satsuma lords, but they are not.

I know Honda makes cars

This is not the same Honda as the Sengoku-era clan you are thinking of. The car manufacturer is 本田, the Sengoku-era family (for example, Honda Tadakatsu) is 本多.

and Hiroyuki Sanada is an actor

He is almost certainly not related to the Sanada clan at all as Sanada is his adopted stage name, with his real surname being Shimosawa.

Odds are that he's not from a bushi family with that surname -- most geographical surnames like that are indicative of a common origin with the surname being granted by a local priest or official. I did some searching in Japanese to try to see whether or not it was a bushi name, including looking for a family crest, but found nothing.

It's impossible to prove a negative, but the evidence seems to be against the Shimosawa being a cadet branch of any of the major families.

Some of these names are still in common use today, but those families are often not related or only distantly related to the original clan.

Common names still in use are: Date, Oda, Imagawa, Mori, Otomo, Uesugi, Takeda, and Hattori.