Does this mean that if one (or both) parents have something major missing (like a limb), the kid will come out without a limb?
Unless there is a genetic basis to the missing limb, no. Even for people born with missing limbs, this is usually not caused by genetics, but by a problem during limb formation in the womb (for example, through medication taken by the mother). A genetic condition associated with missing limbs is tetra-amelia, which is indeed heritable; however, it is very rare, a recessive trait and people actually showing it rarely have children.
If someone got melanoma from laying in the sun A LOT, would their kid have a higher risk of getting it?
No, at least not in the way you seem to think from the way you phrased this. There are people who are more susceptible to melanoma due to genetics. For these people, their offspring can also be more susceptible to skin cancer. However, this is passed to the next generation whether or not the parent has melanoma.
DNA can change over time
Yes, it can. DNA even in adults can change over time, but either this is your germline cells, so it will be passed to the next generation, but won't bother you, or it's in the rest of your cells and may or may not have an effect on you, but won't be passed to the next generation.