Thank you for your rapid and point-by-point response.
An Arduino Uno should handle this easily. I asked because some Arduinos have little memory, and some Arduinos might not handle the pins that you specified. You are OK.
I asked about code because usually people come to this forum with some code that has issues. With no code, perhaps you want the "Gigs and Collaborations" section of the forum. I don't want to write code for you (or for anyone), and it would take away the learning experience that an Arduino is supposed to give to you.
The relay module picture provided little information, but it appears to be optically isolated, with transistors driving the relays and with diodes handling inductive kickback from the relay coils. Good!
Unfortunately, there is no information about current draw for the relays and their driver board, and no information about current draw for the PIR. I am concerned about drawing too much current from the Arduino's power supply. I don't know what the Arduino's limit is, but it is low.
If you study and learn about Finite State Machines, then this project should be easy. The FSM will also be a good way to avoid the delay(...) function which is evil and is to be avoided. Actually, for what you specified, delay(...) could work quite well BUT eventually you will want to do some additional things; delay(...) will interfere with these things and cause you to write your code in an undesirable way. Instead, learn to use the millis() function and unsigned long variables.
Good Luck!