No. The group your user belongs to is irrelevant if the file or directory is already owned by your user. The group permissions are for the other members of the group which owns the file or directory.
I’ve written an elaborate tutorial about permissions, and I recommend reading it. 
The discard mount option has nothing to do with when the system is booted. The discard mount option tells the device to issue a discard operation every time a block is changed on disk.
Not only does this slow down the writing if the underlying device does not support a queued trim, but the instruction may actually be ignored by the device if the amount of cells that would be cleared isn’t big enough.
Therefore, running a weekly trim by way of fstrim.timer is still the recommended way.