If Vader thinks he killed Padme, why isn't he more surprised that he has children?

At that point, there's VERY little left of Anakin in him. He has no memory or love of Padme left, just the Dark Side of the Force

It's possible that you don't witness the time when he actually discovers this. Rather you only see the times after. Also because of his personality change he doesn't feel the natural reaction to meeting them.

I have this image though of Vader secretly going to see Luke when he was young, hiding just to catch a glimpse of him, then leaving.

The simple answer..."the Force is strong in this one." Anakin/Vader knows his strength from his arrogant youth and knows his weaknesses from his 15+ years of servitude, (additionally his regrets.)

As for Padme, she did gasp after he stopped choking her. But regardless, I believe he knew she died but was unaware she gave (premature?) birth of her children prior to that. Clearly her death was known by Yoda and his interest and (Jimmy Smits) was to not hide it.

As to whether the Force should be strong enough to tell if her body still held his unborn children...I am not strong enough to tell.

Out-of-universe, it's because the death of Padme hadn't been planned out when the original scenes were filmed - IIRC the original plan was that Luke & Leia's mom had run off to seek shelter with one of her friends, namely Bail Organa of Alderaan (which also explained how Leia could have memories of their mother).

In-universe, we can say that Vader sensed no deception on the part of The Emperor, and therefore assumed he was telling the truth about Padme's death. The death of the child (Vader knew she had been pregnant, just not with twins), on the other hand, was never mentioned. Most likely, Vader just assumed the unborn child was dead as well. Later, upon learning the identity of the pilot who blew up the Death Star, he put two and two together and assumed that Obi-Wan - who was last seen leaving with Padme while Anakin burned on Mustafar - was able to save the child, but not Padme.

It's what napoleon wilson says: A good alternative to the obvious "because when he finds out, he didn't know 20 years later somebody would come up with a backstory where he's not supposed to know he has children".

When George Lucas made the first Star Wars movie he didn't even intended that Darth Vader was Luke and Leia's father. He came up with that later in the proces when the movie became such an unexpected hit.