Are sequencing reads written differently when inserts and indexes are sequenced seperately?
The order of sequencing in a dual index flow cell is:

Read 1
Index 1 (I7)
Index 2 (I5)
Read 2

Between each step there is a switch of the primer used (Read1-primer -&gt; Index1-primer (I7) -&gt; Index2-primer (I5) -&gt; Read2-primer)
There are differences in how the I5 is dealt with between dual- and single-read flow cells as explained in the Illumina Indexed Sequencing Overview. In this document there is a lot more detailed explanation of the sequencing workflow that may be helpful to you.
But as explained by the other answers, none of this will matter in most cases. Even if you decide to output the Index sequences they will be exported as separate FASTQ files.