How is the prostration of forgetfulness (sujūd as-sahw) performed in detail? Does it include the takbīr and the specific dhikr for pro prostration and rising from it, followed by the taslīm immediately, or is it necessary to recite the first and second Tashahhud before the taslīm?

The description of the sahv prostration (prostration of forgetfulness) is correct and sufficient. The prescribed ذكر (remembrance/supplication) for it is the same as that for the prostration during regular prayer. As for the tashahhud (testification of faith) after the two prostrations of sahv, there is a difference of opinion. If its prescription is affirmed, it should be short (only the first tashahhud). And whoever doubts having omitted an obligatory part of the prayer must perform the sahv prostration.