they can study the genetics of modern organisms because they can extract genetic material from them.... but what genetic material can they extract from fossils? Zilch.
True. But... bones have features that are guided by genes. Those features can be studied and compared. For instance, the suborbital constriction on the skull, or the sutures, or the dental arc, or number and shapes of cusps, or ...
When two very similar species have many shared features but some different, and then you have a third species that share features with the second one but not the first one, and so on, then you know they are related. It's like how the textual comparison of the Biblical fragments are done to date and compare which one was copied from which. If this doesn't work for evolution and the fossil record... well, then it doesn't work for Biblical historical research either.
So yes its true that living things can be traced, its not true that fossils can be traced.
Yes, they can. The bones, both in shape and composition (you can study trace elements and chemical composition to find out diet and other things), can tell more than you think.
Secondly, the fossils are organized in layers that are of different ages. It doesn't matter if you believe in dating methods or not, the deeper strata is older strata, and older strata contains simpler lifeforms and simpler forms of the newer species. It's like looking at a movie, picture by picture, still frame by still frame, and see how it slowly changes to simpler forms.
And likewise, observing evolution in organisms with short generation times only shows how those organisms can adapt over a short time. You dont see those organisms change from an organism of one phylum into a completely different phylum. That has never been observed.
The adaptation that you're talking about is genetic change. The shape of a person, being, species is defined by the genes (hox genes and such). And those genes do mutate as well. So if the genes that controls your body shape can mutate and do mutate, then those mutations cause now body shapes. Simple. And this has been studied too. Change hox genes in a fly and you can give them extra legs or wings. Change them in a chicken and you can give them teeth (they have the gene for teeth, but they're dormant=turned off).