Is it possible to derive the encryption method from encrypted text? Is it possible to identify the encryption method, or at least rule out some of them, by looking at the encrypted text?

For example, if you have 3 encrypted strings where the first 10 characters are the same on each of the strings.
If the encryption system is any good, then no. The output of a (symmetric) encryption algorithm is supposed to be indistinguishable from pure random. If you can distinguish encryption output from pure random with probability greater than 1/2 (i.e. you are given two strings of bits, and you can tell which one comes from an encryption system with better success on average than pure luck), then the encryption algorithm is said to be "broken". An example of such an algorithm is RC4 (not that being broken does not mean that decryption without the key is easy; RC4-protected SSL connections still appear to be mostly safe).

However, the used encryption algorithm is not generally supposed to be secret information. That algorithm exists as code or specific hardware somewhere; it is amenable to reverse engineering. It is very difficult to know how much algorithm secrecy may resist an attacker, and that's a problem since a good security policy should aim at quantifying things. Secrecy of a symmetric key can be quantified: a 128-bit key is secret up to 2127 (on average) trials. Since such a precise measure cannot be achieved for the algorithm itself, it is usually assumed that the name of the algorithm is known by everybody. Correspondingly, encrypted message formats often begin with an additional header which plainly states which algorithm is used (such a header avoids many maintenance headaches, especially when several distinct algorithms are in use).