Ivy, yes. Easily. My favorite way is to cut a long piece and bury it in a pot in a circle with only the leaves above the soil. Very shortly you can dig a little and see the roots forming on the jointed parts. You can then cut those to individual pieces or just leave it to grow. I've also done this with the plant still in the ground and just curled a branch into the pot on the ground right by it. Cut off when it has developed roots.
Fruit is a different matter. Most producing fruit trees are grafted onto different rootstock for disease resistance and other properties. I never did much studying on that subject. I do know that the fruit trees I've grown from
seeds never produce fruit that is worth much either size-wise or flavor-wise.