So with my basic understanding of biology, DNA can change over time.
Does this mean that if one (or both) parents have something major missing (like a limb), the kid will come out without a limb? Does it matter if the parents had limbs amputated or were born like that? If someone got melanoma from laying in the sun a lot, would their kid have a higher risk of getting it?
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.
DNA can change over time
This is a terribly misleading thing to say. You don't have "your DNA", you have a few billion copies of your DNA, one for each cell 1. Within an individual, there are events (called mutagenic) which can change the DNA in an individual cell. The change is practically random. It is like having a pile of DVDs and tossing them around, so some of them get scratched a bit. If one event hits many cells at once (for example a strong dose of ionising radiation), the cells will likely get different mutations each. There is no event during adulthood which will lead to having the same mutation across all DNA copies of an adult organism. 2
A second point: most events which change something in our body don't change a single copy of our DNA. Amputation of a limb is such an event. All (remaining) cells of an amputee have the same DNA as before.
Having cleared up that, the implications for your question are simple, YviDe already mentioned them.
If two humans experience a non-mutagenic event in their adulthood, their DNA will be the same as before, and the consequences of the event will not be inherited. Thus the child of two people with amputations will be born with all limbs present.
If two humans experience a mutagenic event in their adulthood, it will change the DNA in a very small part of their cells. If they get a melanoma due to UV damage, it's only the DNA in the tumor cells that's different (and probably the DNA in other skin cells which got changed, but not enough to turn them into tumor cells). The DNA copies in the rest of their body, including the DNA of their egg cells and sperm cells, stays the same. Therefore, their child is born without a melanoma.
If two people were born with the same mutation which causes a visible symptom like a missing limb, their child will be born with it too, and will also be limbless. In this case, parents and child have the exact same mutation in all cells throughout their bodies. This is however not due to an event which "changed their DNA over time". Also note that not all birth defects are genetic.
1 This is not exactly true, but true enough for this explanation.
2 I shortly pondered whether a retrovirus can do it across the body, but all the ones I know of only work on certain cell types.
Does this mean that if one (or both) parents have something major missing (like a limb), the kid will come out without a limb?
If the parents are born without a limb, it could have a genetic reason or from external factors such as medicines and radiation. If the baby was born with gene mutations, he may come out without a limb.
So if mutations have taken place in the parents, they may have the chance to be passed on to following generations. And the condition you have presented may be caused by mutations. This is also suitable for you second question.
