Do you think cucumbers and melons would grow fast and big?
again you "can" (and I say that loosely).. grow them indoors.. but really be prepared to have enough heat and light to cover them fully.. and I would use some type of fake grass under them so they are not sitting on the floor itself.. also would use stakes/poles or something because they get viney.
I started cucumbers, zucchini and pickles indoors.. took awhile for them to get going but managed them - they just started getting too much for me indoors to deal with.. so unfortunately I had to end the process on those. atm I have a purple bean (green bean but purple), growing indoors.. it started flowering.. if I'm lucky I'll get 12 - 15 beans from it.
but to answer your question, they will grow... maybe not so fast and so big.. again that goes right to what I said above about providing the right amount of conditions. I would say you would have more luck with the cucumbers over melons indoors tho.