Hint: my intended answer is an anagram of this one.
That leaves
TOTO
If it was “Cycle back and forth for dog” I might have gotten it.
Sounds like Fiddler star and an original gangsta are into deformed geometric objects. (8)
It was actually OTTO, as in the Otto cycle for internal combustion engines. But never mind, carry on 
.
TOPOLOGY (Topol + O.G.)
Whether you end up last or with a large boat at the end of the alphabet, it’s all in how the dice roll. (7)
Yahtzee
++++
Strange that in the end the Riche wear spaghetti sauce. (5)
EERIE (last letters)
Approximation method ruins a summer inn. ((7, 3)
RIEMANN SUM (anagram)
Sounds like the largest member of the deer family ate a light and airy dessert.(6)
Mousse
+++
Northeastern influences in Asian religion intrigue OG YA writer (1,1,6)
S. E. HINTON (Shinto + NE)
Hairstyle causes nurse to panic. (7)
TONSURE (anagram of “nurse to”)
+++++
Brief class on large container found by Nevada city, remodelled (9)
RENOVATED (Reno+vat+ed)
++++++
Dilapidated relic Ted broke.(8)
DERELICT (Ted relic*)
+++++++
Determine hall is contaminated here! (5, 4, 6)
Anagram, if hint needed.
I get that it’s an anagram, but there’s a lot of possibilities. The best I can think of is THEIR ILLS DEMEAN.
The clue ends in a !
In cryptics this means that the clue in total works as both definition AND wordplay. So where might you determine a hall is contaminated?
Aha - THREE MILE ISLAND. Nice - I too had already figured it was an anagram, just couldn’t find the right words.
+++++
Darned departed pet (7)
I thought this one would be easy, as it’s much more of an Americanism than anything else. And while “departed pet” could describe my username, my username has nothing to do with the answer - just a nice coincidence.
Doggone
+++++
Campbell correct, then Hell freezes over (5)
NEVER (Neve+R)
++++
Passenger refused the seat on the right.(9)
“refused” and “seat on the right” are the clue elements.