This sounds like a mechanical fault, the water in the overflow under where you place the glass is most likely caued by a failing valve.
The issue where ice isnt dropping into the container is likely due to the mechanism becoming stuck, a defrost would normally resolve this issue, or failing that a new actuator may be required.
Since there is at least one hardware failure, I would suggest you get an engineer to site to strip down the door where the two components are located, and replace the faulty water valve. and then inspect if the ice mechanism is stuck or actually needs replacing.
If you would like me to recommend a suitable engineer, please advise your post code.
I would recommend DAW, we have recommended them in the past and have always had good feedback from the customers who have used them. From looking at the post code they must be just up the road from you which is a bonus.
They charge a £40 call out fee which includes labour repair cost. Its then just the cost of the parts required, which I dont think will be overly expensive.
They can be contacted on 01253 590380
The cost of parts is something which is controlled by Samsung, unfortunately identifying the exact parts which needs replacing is something which can only be done on site, I did my best to rule out something which would have been a wasted visit (in terms of something you could do to self fix), and also found an engineer who offers a very competitive rate.
If you didnt use the service you could have run the risk of calling someone out and it being something you could fix yourself, in this instance once the unit was inspected multiple parts were deemed faulty and this meant it was uneconomical to repair.
I did my best to rule out any unnecessary costs, but you are now in a position to know the fault would be cost prohibitive to repair.