1) I recently had my apartment floor done with floating laminates. The contractor hired proved to be very inept in doing the job and has shoddy workmanship. I want to know what people should do in typical situations to install quarter rounds properly along the baseboards near door jambs.

2) I am trying to get him to redo his work or bringing it someone else to redo all the trim work. How much should replacing quarter rounds for a 700 sq ft. space cost??

Other images can be found here:
http://www.flickr.com/photos/40028963@N05/

Thanks.

Ouch!
First mistake. Using stained trim that does not match the floor. The 1/4 round should have been the color of the trim already in place.

The fix is to remove the trim. Throw it away.
Then take the existing trim and take it off. This colonial trim that is in place is not square on the face. Or so it appears. You cannot put any trim against a un-even surface and expect it to look good. What the contractor should have done is to either undercut all of the trim as they did at the door way and slid the flooring under. Or removed all of the trim, undercut the door jambs. And then replaced the trim after the floor was placed down.

There is no other way around this if you want it to look ok.

What I see from that picture is they did not stop the new floor in the middle of the door jam. There's a gap in the flooring, that's not 1/4 round, it looks like cove moulding, or it's some funky form of ogee installed upside down it should have been placed between the two door cases not over them, it also needed to be back cut at a 45 deg. angle not streight cut, it should never have been what looks like the cheap looking peel and stick wood grained fake wood moulding. It should have been real hardwood stained and sealed to match the flooring or simple paint grade and painted t match the trim as suggested.
If I had of seen one of my employees do a job like this I would have sent him or her home ripped it out myself and redone it at no cost before you ever saw it, (hope fully).
If the boss saw this and has no idea why your making a big deal of it, then there blind and stupid.
That whole job was a carpentry 101 job. I hope you had no future plans to use them again.
I took a look at the other pictures you down loaded and without a doubt that's the worst trim job I have ever seen. I would never allow them back on the property.
One last thing, was that the origanal baseboards?
It looks like someone used casing instead of baseboard. What that does is make the baseboards even with the door casing. The base boards should have been thinner so there was a step to end the 1/4 rds.

I understand your point exactly and that's why I was totally shocked. I have had someone redo the shoe molding in white rather than taking out the whole base boards which would have delayed my move-in date and just leave it at that. The baseboards are the original, probably 15 years old or so, maybe that's just the way they came back in the day. Obviously I have no intention to use them again and will warn everyone else against using the same company. However, since the owner WAS the installer in this case...he didn't see anything wrong with his work.