Your information is a bit confusing. Your photo is fuzzy but it does not show a Radiotronix RCT-433-AS. Your schematic diagram shows the RCT-433-AS, so I'm wondering what you think the relationship is between the two. In any case, it is safe to presume that the manufacturer of the module that you have has designed it carefully and that you should not be desoldering anything from the board.
The components between the IC and the antenna (the white coil of wire in your case is the antenna) are there to match the antenna impedance to the requirements of the IC output. If there are resistors, such as is shown in the schematic for example, they are in place to form a T-pad to buffer or isolate the antenna impedance from the IC output so that the IC does not malfunction. If your own board does not have these resistors, don't worry about it, this is a sign that they are not needed on your board. These resistors are needed by some transmitter ICs but not others.
Your range will depend on the antenna that is on the receiver, but you might expect to reach 50 feet.