Undermount sinks are extremely common and if done properly, have no more issues than top mount sinks. They are an additional upcharge from the counter fabricator because the cutout has to be polished, and the sink attached correctly. The charge usually is from $200-$300 for that service, in addition to the more costly sink. Installation requires clips, brackets, or a cradle to hold the sink to the counter. Only hacks just "glue" the sink to the counter without additional support, and that perhaps is the reason the builder has had negative experiences with them.

As far as replacing the sink down the road, yes, it's possible to replace the sink with the exact same model as long as the sink cabinet itself is actually large enough to accommodate the sink you choose. I actually had a client whose husband had a firearms accident that shattered a silgranite sink on an install less than 6 months old. It required the sink to be replaced, and it wasn't a big deal at all.

What is the counter material?

Anything that can be damaged by long term water exposure is an issue.