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12. Laborious stuff, but works

• $\#_1$

The following:

(Transpose[{list, Values@rules}] /. {x, i_} -> {i, i} /. {x^2, 
  i_} :> {i^2, i} /. {x^3, i_} :> {i^3, i})[[All, 1]]

and

• $\#_2$

equivalently

(Thread[{list, Values@rules}] /. {x, i_} -> {i, i} /. {x^2, 
  i_} :> {i^2, i} /. {x^3, i_} :> {i^3, i})[[All, 1]]

7. While? also works works

Needs@"GeneralUtilities`"
Module[{hold},
  SetAttributes[hold, HoldAll];
  
  oneTimeRules[rules_] :=
   
   Normal@Merge[rules, ListIterator] /. Rule -> RuleDelayed /. 
     i_GeneralUtilities`Iterator :> 
      With[{r = Read[i]}, hold[r, r =!= IteratorExhausted]] /. 
    hold -> Condition;
  
  ];
 
Replace[list, oneTimeRules@rules, Length@rules]

7. While also works

Needs@"GeneralUtilities`"
Module[{hold},
  SetAttributes[hold, HoldAll];
  
  oneTimeRules[rules_] :=
   
   Normal@Merge[rules, ListIterator] /. Rule -> RuleDelayed /. 
     i_GeneralUtilities`Iterator :> 
      With[{r = Read[i]}, hold[r, r =!= IteratorExhausted]] /. 
    hold -> Condition;
  
  ];

and then

Replace[list, oneTimeRules@rules, Length@rules]

11. Another fancy approach based on Michael E2's answer. This serves as a clarifying comment. Grab the relevant code that is needed

SetAttributes[useRepeated, Listable];
useRepeated[(Rule | RuleDelayed)[pat_, repl_], n_ : 1] :=
  Module[{used = 0},
   pat :> repl /; used++ < n
   ];
useOnce[r_] := useRepeated[r];

and then use

Replace[list, useOnce@rules, Length@rules]

or equivalently

ReplaceAll[list, useOnce@rules]

Note: if one uses Replace and the default value of the command -which is set to 1- the above does not work. So some minor caution is needed.