Signatures and Fingerprints
Simply put, do PGP signatures provide file integrity verification in addition to file identity verification?
Yes, they do: but without further investigation, they only verify whether a given key was used to sign; not whom it belongs to (read on below)!
If I've verified a file's PGP signature, are the MD5 and SHA1 sums of the file more or less irrelevant, as I've already verified integrity?
Yes, unless you haven't been able to verify the ownership of the key, but the other person provided you with the correct fingerprints in a secure channel (by meeting, (video) phone call, ...).
If I've verified a file's PGP signature, are the MD5 and SHA1 sums of the file more or less irrelevant, as I've already verified integrity?
In fact, OpenPGP also uses hashing algorithms like these (whereas it hopefully shouldn't use MD5 any more, which is considered too weak our days). But digital signing adds information on who signed the software (if verified correctly).
Verifying Key Ownership
Unless you directly signed the software issuer's key, all you know is somebody signed the software. There is no primary certificate authority in OpenPGP, everybody can add a key with arbitrary names, it is up to you to decide which to put trust in. This is usually done by meeting and exchanging the key's fingerprints, but depending on your level of paranoia you might be fine with verifying the key fingerprint against some they put on their website (which then should be transmitted SSL-encrypted together with a reasonable certificate, of course).
Then, there is the concept of the web of trust, which allows to build trust paths from you to the issuer without meeting him directly, by trusting others in-between that together form a chain of certifications (this can actually be compared to the intermediate certificate authorities of X.509). A very short explanation can be found in this answer: What is the exact meaning of this gpg output regarding trust?.