Do you think Cell would have given Goku and Gohan a chance to get stronger?

[Sensiblesaiyans]

Let's suppose in an alternate timeline, Goku and Gohan went into the ROSAT first and faced semi-perfect cell. Due to a chain of unforeseen events, Cell manages to become perfect and fights Goku followed by Gohan. Now assuming both were defeated by him, can you see Cell being merciful to them, giving them more time to train and announcing the cell games?

[Vegeta th3 4th]

I assume he would, as he did so for Trunks and Vegeta. In fact, I'll go a step further and say he may have done it in the Cell games if no one could beat him, assuming he enjoyed his time.

[ScouterSSJ]

In that context, yes, without any doubt, in the same way he gave it to Vegeta and Trunks.
Cell only became fixated in killing everyone at any cost once he was totally surpassed by Gohan and his SSJ2, but in a context where he doesn't unlock that form and is defeated, I could see it.
Now that being said, had Goku and Gohan went in the RoSAT before Vegeta they would've come out of it as strong as they did in the series, and since Goku had a good idea of Cell's power thanks to his fight with Frieza and his plan always was to make Gohan evolve during his fight with Cell, I don't think that scenario could ever happen.

Goku would've let Cell complete himself no doubts (he in fact did when he didn't help Piccolo kill him when they were looking for him on a plane), but even if they were fighting alone, Cell would've no doubt teased Gohan until the boy would've cracked, and so the same scenario of Gohan trashing Cell and Cell wanting to kill them all would've repeated itself (me thinks).

[Zephyr]

Yeah, Cell wants to have a fun fight, and he's open up the possibility of endangering himself in the process.

[Peach]

Goku after he resigned, yes.

After that? I'm unsure. He was willing to resort to cheap tactics like self-destructing.