Dem Bosain

Sorry, none of those are true.

-GIMP is freeware.

-It’s also open source.

-It’s also free to distribute.

-It’s free to modify.

-It’s even free to sell, as long as you include the source code.

Please explain. How is FOSS not also freeware?

Without that extra loop, it’s possible that flushing the toilet will suck the water out of your sink trap.

Ignoring scale and any local requirements, yes, that’s a valid trap and vent.

Be aware that local inspectors can be touchy about t-junctions. You might need a rounded or angled junction here so the new drain doesn’t splash directly against the opposite side of the vent.