Is Bilbo really Frodo's "uncle"?
No. Bilbo is the son of Bungo (son of Mungo). Frodo is the son of Drogo, who is the son of Fosco (son of Largo, Mungo's younger brother). Therefore Bilbo and Frodo share a common ancestor in Balbo, father of Largo and Mungo. Since that makes Bilbo and Drogo second cousins, Frodo is Bilbo's second cousin once removed.
And (I almost forgot)--Frodo's mother Primula Baggins (nee Brandybuck) is a first cousin of Bilbo on his mother's side. Hence, through their Took connections (Primula is the daughter of Mirabella Brandybuck (nee Took)), Bilbo and Frodo are first cousins, once removed.
Are Merry and Pippin actually cousins?
Yes--they are first cousins, sharing a common grandfather: Adalgrim Took (who, by the way, was not a Thain).
Merry and Pippin are first cousins, twice removed, of Bilbo (via Bungo's marriage to Belladonna). They are also Bilbo's second cousins, twice removed (via Rosa Baggins' marriage to Hildigrim Took).
Are Merry and Pippin related in any other way?
Yes. Merry's great-grandmother, Mirabella Brandybuck, is Pippin's (and Merry's) great-great aunt. Therefore, on Merry's father's side, Merry and Pippin are also third cousins (sharing the OldTook? as a great-great-grandfather).
How is Frodo related to Pippin?
In the same respect as Bilbo is related to Frodo, as it turns out. Frodo and Pippin share a common ancestor in the OldTook?, who is one of Frodo's maternal great-grandfathers. This makes Frodo and Paladin second-cousins; and Pippin is therefore Frodo's second cousin, once removed.
How is Frodo related to Merry?
Frodo and Merry share a common ancestor in Gorbadoc Brandybuck, who is Frodo's maternal grandfather. Frodo is thus a first cousin of Saradoc, making Frodo and Merry first cousins once removed.
However, Frodo is also connected to Merry on the side of his (Frodo's) maternal grandmother, Mirabella Brandybuck (nee Took). Frodo and Merry share a common ancestor in the OldTook--and? since Merry and Pippin are first-cousins (sharing a Took grandfather), Merry is also Frodo's second cousin once removed.
Is Frodo related to Merry and Pippin in any other way?
Yes. Rosa Baggins, first cousin of Fosco, Frodo's Baggins grandfather, married Hildigrim Took. Via this connection, Frodo is a third cousin, once removed of Merry and Pippin.
Where does Fatty Bolger fit into this picture?
We can only be certain of Fatty's links to the other hobbits via the branches of the Took family tree published in The Lord of the Rings. There we see that Fatty, Merry and Pippin share a common great-great-grandfather in the OldTook?, so Fatty is a third cousin of Merry and Pippin. Fatty's mother, Rosamunda Bolger (nee Took), is Frodo's second cousin, making Fatty Frodo's second cousin once removed. Fatty is Bilbo's first-cousin, twice removed.
The Baggins and Brandybuck family trees indicate other Bolger connections, so Fatty may be related to the others in additional respects. And let's not forget the fact that Fatty later becomes Merry's brother-in-law.
And what about poor Sam?
Sam is, of course, not blood-related to the wealthier Brandybuck, Took and Baggins families (if he is, the connections are lost in the mists of time). It is not recorded if his daughter Goldilocks and her husband Faramir the Thain produced any children; if they did, such a child would be:
Pippin's grandchild; Merry's first cousin twice removed (and third cousin twice removed); Fatty's third cousin twice removed; Frodo's second cousin thrice removed (and third cousin thrice removed); and Bilbo's first cousin four times removed (and ''second cousin four times removed) (that should give you an indication of how long-lived Bilbo was).
A regular warren, by all accounts!
How does one establish who is related to whom and in what degree?
If two people share a common ancestor, this makes them blood-related. Merry and Pippin are first cousins, for example, because they share a grandfather, Adalgrim. Bilbo is Adalgrim's first cousin, so Bilbo is Pippin's first cousin, twice removed (i.e. because he is "removed" from his grandfather by two generations). Bilbo shares a common ancestor with Pippin in the OldTook?, who is Bilbo's maternal grandfather, and Pippin's paternal great-great grandfather. Bilbo and Drogo are second cousins, because they share a great-grandfather (Balbo Baggins). And, as mentioned above, Fatty and Pippin are third cousins, because they share a great-great-grandfather (the OldTook?).