Error Correcting Code Generator Circuit Implementation with XOR gates
This question is quite open ended but ECC was fun to explain to my other friends back in college so where we go. Firstly, a correction: This produces a 12bit number, not a 13bit one. Data bits 0-7 and parity bits 0-3. Your schematic is wrong and would never work.

"If I feed inputs 3 5 7 9 11 into an XOR gate, will it indeed give me the result of the parity bit 1 (P1)?" Yes. That xor gate must have 5 inputs but yes. Also you shouldn't feed the outputs 3,5,7,9,11 of your circuit to the input so please refer to the equation as P1= D0 XOR D1 XOR D3 XOR D4 XOR D6.

If you don't have an XOR gate with 5 inputs then you'll need to ask a new question telling us what you actually have to work with.