I have about 40K through a lee 4 hole, almost all of them using the auto index, and know the press and its personality pretty well. I will give what advice I can.
If SPP primers don't seat correctly in .357, but do in other cases, its most likely the brass. It could also be the shell holder. The retracting collar may be jammed on the shell holder.
Primers on the floor are usually a product of the safety primer being too high or too low. If this is not an issue with LPP, most likely your decapping die needs adjusting. The decapping die controls the primer arm position and relative height to the safety prime. With the 4 holes, each die does not need to line up with the others, so this shouldn't be a problem.
As annoying as it sounds, with longer cases, and shorter or looser cases, you have to get a feel and guide the case into the sizer. This is very easy once your in the habit, and can still allow good speed. I do this and run about 180 rounds per hour. I need to do this with .357, and .380. Just the way it is. The nice thing, you sometimes have to hold pressure on the case to get them to prime straight (maybe part of your above issue as well?) and can keep your thumb in position to put light pressure on the case going into the die, and priming. Again- I know this is awkward, but it pays off.
The auto drum.... I'll get to that at the end, but long story short, get an autodisk, or whatever replaced it. The auto drum is not a great measure in general.
The clunkyness of seating and crimping SCREAM to me that your not flaring the case enough when charging. This is an easy adjustment, and will indeed result in that. It can also ruin brass.
That scraping going into the seating die should be the flair getting pressed out from the charging station. Maybe its something else, but there is always a scraping drag on seating flared cases. I'm guessing you don't have enough flare to get the bullet in smoothly, but enough to feel/hear resistance.
Most importantly with the Lee auto indexes, replace the square ratchet when its worn out. They cost .50-2.00$. Once you have the motions figured out, they last a very long time. I'm about 20,000 rounds on my current ratchet. My first lasted 300 rounds -learning curve. Second lasted a few thousand. Third lasted many thousands, 4th is current.
NEVER half stroke. Always all the way up, all the way down.
Do not be afraid to put a wrench on the index rod, and turn the die turret. Sometimes those turrets just don't line up, and that nut on the top of the index rod is adjustable. Its very tight, but is adjustable. Those adjustment are why my first ratchet lasted 300 (I didn't know at the time. Lee instructions didn't core it), and current is on 20K.
Sometimes the weather effects how they run. I have found with high humidity but no rain, my press gets finicky. Runs great when its raining, runs great in low humidity. This isn't superstition, the Lee's use a couple Nylon parts in the index system, and Nylon absorbs water and acts funny. Maybe I'm exaggerating in my mind, but there is something there. When it does that, I can make more adjustments, or load another day.
On to the auto drum....
So I got a free auto drum powder measure, and WOW! was it bad. I use it now, but It took work. I'll tell you what I did, but can't say you should.
1: The auto disconnector is junk and shouldn't be used. I don't think you can anyway on a 4 hole. You put the screw into plastic part to lock it out. I talked to Lee.... they pretty much agree.
2: Some powders will leak. No way around it.
3: The threads on mine only engaged for 4 threads. Way to little for aluminum, with a 70lb cycle load. This was a product of trying to make the measure "universal" AKA sucks at everything equally. I fixed this in my application by cutting down the internal drop tube to where I could get it threaded a reasonable amount. This dramatically improved performance. It also limited its ability to charge other cartridges. I used a standard cheap pipe cutting tool to get it square.
4. The plastic inserts are "as-cast". I put the screw shaft in an electric drill and turned it with 800 grip paper. After that, ran a few dozen charges to get the graphite coating. Leaks dropped 99% And it did leak before. Around 10% of powder dropped leaked. Yes really. its sucked that bad. 25 grains in .223=3 grains on top of the die. This combined with better die engagement took me from 2-3 grains off per charge -Yea definitely in the dangerous range to now in the 0.2 range. Well within safe if your not running max loads.
5: I don't know if you have this problem, but mine did. Some do, some don't. The above mentioned 70lb cycling. The charge return spring on my came in at appx 70lbs force to compress. This buckled many 223 cases. Straight wall would do better. I took the measure apart and crushed the spring. This can be extremely dangerous so I won't go into it, but this took the spring down to about 15 lbs of force, similar to a normal measure. If you have this problem and a safe way to stress the spring down to normal, it may be worth it. I have seen people grind them to weaken them, but I didn't like the idea.
I hope some of this helps, and feel free to PM me for clarity, or any question. I know this board has many experienced with this press, and I'm sure some of your questions have been answered, but I wanted to address all of your issues from my perspective.