Let me preface this question by saying that the Czech Republic and Chechnya are two different countries.
Are the two countries' names etymologically related, like Austria and Australia are?
Wiktionary doesn't have the etymology for Chechnya, nor does the Online Etymology Dictionary.
There is no apparent relation.
The English name Chechnya is a transliteration of the Russian (Чечня́), which according to Amjad Jaimoukha's The Chechens is a 17th-century Russification of Shashan, a Kabardian name for the people who call themselves Nokhchiy (which may be derived from the words for sheep cheese, the plow, or the Biblical Noah). The traditional etymology links it to the village of Chechen-aul, where the Russians were defeated in battle, but the name predates the battle.
The Czech Republic takes its name via Polish from the Czech tribe, which conquered the region in the 9th century. That name in turn may derive ultimately from četa, a military unit in several West Slavic languages. It dates from the 18th century; the traditional English name was Bohemia, used into the 20th century. The traditional etymology links Czech to Čech, the legendary founder of the country.
That was a great explanation! I want to comment on another matter mentioned in the question. The names Austria and Australia are totally UNrelated. Austria comes from German Österreich that cognates from Old High German Ostarrîchi (996) or Osterrîche (998) [E. Zöllner, Der Österreichbegriff, 1988], from Medieval Latin Marchia Orientalis, an eastern prefecture for the Duchy of Bavaria established in 976.
Australia means "Southern Land" in New Latin, adapted from the legendary pseudo-geographical Terra Australis Incognita ("Unknown Southern Land") dating back to the Roman era. First appearing as a corruption of the Spanish name for an island in Vanuatu in 1625,[61] "Australia" was slowly popularized following the advocacy of the British explorer Matthew Flinders in his 1814 description of his circumnavigation of the island.[62] Lachlan Macquarie, a Governor of New South Wales, used the word in his dispatches to England and recommended it be formally adopted by the Colonial Office in 1817
