Yes. When eggs are first laid they'll be soft but fertile eggs will firm up within a few hours. They can then be moved for incubation. Make sure to mark the top of the egg with a line so you don't rotate the eggs.
I just use my hands. Fertile eggs almost always for me at least firm up almost immeadately. Infertiles seem like while white the shell is missing . . .
I just use fingers. I've tried the spoon technique, but it doesn't make me feel like i have good control over the eggs. They'll be a little firm unless you catch the gecko laying them, and in that case they'll be squeeshy. I'd let them sit for a minute to firm up, if you catch her in the act.