This is follow-on to earlier post with more info as I have a picture of pipes behind my bathroom wall. The basement sink drains very slow after an earlier sewer backup into the toilet which was cleared by a plumber's snake. I would like to fix the sink myself, everything else in house drains fine. I earlier tried a plunger and a 1/4" drum auger with regular head. The auger appeared to go way up a pipe instead of down to clog. I forgot about this opening where I could see the pipes (barely).

The white PVC is the sink drain and it enters a vertical drain at a downward angle (so it seems). Going up leads to ABS vent piping (it connects above to the vertical ABS pipe closest to me). Based on noises I heard doing the auger the cable seemed to have somehow gone up. Maybe the clog is right at the junction which forced it up? It may have been at that point where it would only let me feed the cable a couple inches at a time before I had to rotate it, maybe it was on this corner where it kept getting stuck going up?

Will a down head/drop head make it more likely to bore into the clog at the junction rather than being deflected up? There is a drum auger from General Wire Spring, a D-25-2 which is a 1/4" x 25ft with down head. Any attachments don't work apparently with 1/4" cable and in order to go through the sink trap starting at 1 1/4" and then going to 1 1/2" pipe it seems I have to stick with 1/4" cable.

As you see in pic the drain goes down then bends 90 deg back with a few feet horizontally until it connects to the 4" main cast iron drain.

that is what a drop head snake is designed for.. it should drop down with no problem.

Is there any risk of the head breaking off? Is it bad if the head is left somewhere in the pipes?

i don't do alot of drain cleaning but of all i have done i have never had a head break off. i have kinked my share of cables..
anyway, you are cleaning plastic pipe so it should be fairly easy.

There is always a slight risk of the head breaking off, and it is ALWAYS bad if it is left inside the pipes.

What is the most likely cause of head breaking off? What if the sewer sludge hardened (it is a drain that has not been filled with water for many days while I figure out what to do)? Will an auger break through anyway or is there a way to soften it first? Should I simply run some hot water first?

Age, or using it in a drain with chemical drain cleaners will damage the snake and make it brittle. THen it will snap off.