οἵ τε λαμπροὶ καὶ λευκοί εἰσιν.
It looks like you are trying to use a τε X καὶ Y pattern, which I don’t think is correct. You could use an X τε καὶ Y pattern instead (and change the relative pronoun to οἱ without accent).
τοῦ κύκνου μακρὸς ἐστίν.
Change to μακρός ἐστιν.
Μικρὸν πλοῖον ἐπι τῆς λίμνης ἔστιν.
Change ἔστιν to ἐστίν.
τὴν αἰτίαν τοῦ αὐτοῦ.
Maybe use τὴν αἰτίαν τούτου (the cause of this)?
Supplementary to katalogon:
ἐν αὐτῷ τῷ πτερῷ is “in the wing itself.” Perhaps this is right, but I’d have expected “in this wing” (ἐν τούτῳ τῷ πτ.). And your following αὐτοῦ τοῦ κύκνου (“of the swan itself’) I expect should be τούτου τοῦ κ. or just τοῦ κ.
If you’re going to do more of these I suggest you give the English, which I don’t have.
A look at Sidgwick’s version—thanks to paveln—confirms our various corrections to the OP’s attempt. But it has an odd mistake. In the third sentence the printed version has οἱ, the definite article. It ought to be οἲ, the relative. These pesky diacritics are a constant source of trouble.
Breathings (rough and smooth) are relatively straightforward (but only relatively).
Accents (acute, grave, and circumflex) are quite complicated, though the basic principles are simple enough. Probert’s Short Guide is the thing to use.
Quantity marks (short and long) are also sometimes found. Any given vowel will be one or the other.
Various other diacritics are used in ancient manuscripts, but are not in modern use.