RSA - Ecrypting the same data with the same public key = same ciphertext? If an adversary knows my public key and guesses what was my plaintext, can he test for it somehow?

The most obvious way is encrypting the guessed plaintext with my public key and the same parameters that he can probably discover from the ciphertext, to see if the result is the same.
The answer to your question is "yes, RSA will encrypt the same cleartext to the same ciphertext every time." And yes, this is a known property of RSA. That's why every standard that uses RSA, such as PKCS or CMS, specifies the use of randomized padding.