The shoulder and arm plates would go over the the jacket. Now, here's my advice if you're dead set on using a trenchcoat: go to a thrift store, buy an inexpensive one and chop the sleeves off and make a trench vest, then the shoulder armor will be on your vest and not the jacket. Not a lot of folks wear them for a number of reasons, mainly they are hard to get to look right and not just a chopped up trench coat tossed on your kit. If I recall correctly, only the trench vest is approved for use and not the full coat.
For the buttons, one member, Ithica, was rather clever. He removed the buttons and replaced them with some fine chain and clasps running across his chest, looked pretty sharp and was a nice little detail.
Now, personal opinion: don't use a leather one. Try to find a fabric one, it will be somewhat lighter and not as hot a leather one. These kits get rather warm, and anything to help you be a little more comfortable is a blessing.