Yes, remember that Tolkein describes the stab by Meriadoc as having an affect on the sinews of his leg. Wraith != ghost.
The Ring calls to the Nazgûl. From LotR, “A Knife in the Dark”:
" ‘They themselves do not see the world of light as we do, but our shapes cast shadows in their minds, which only the noon sun destroys; and in the dark they perceive many signs and forms that are hidden from us: then they are most to be feared. And at all times they smell the blood of living things, desiring and hating it. Senses, to, there are other than sight or smell. We can feel their presence - it troubled our hearts, as soon as we came here, and before we saw them; they feel ours more keenly. Also,’ he added, and his voice sank to a whisper, ‘the Ring draws them.’ " (pg. 202), emphasis added
So the Nazgûl certainly knew the Ring was there, and probably would have figured out who was carrying it given time in the presence of it, even had Frodo not put it on. But when he put it on, it made it trivial for them to figure it out, as he became at that moment visible in their world, on the other side.
And the Nazgûl, while certainly formidable enemies, are not really that powerful arrayed against determined resistence; it’s usually their ability to instill incredible dread in their opponents that helps them be so overwhelming. Certainly the Witchking can swing a mace, but note that his mace merely manages to break Eowyn’s arm, rather than smashing her flat into the ground or some such. Presumably, his brethren wraiths are less powerful than he is. If they could simply overcome physical resistance easily, they would have themselves attacked the Prancing Pony (as Jackson shows happening, though the book makes it clear that the Inn attack is the work of minions, not the wraiths themselves) and taken the Ring then and there. And Strider is not just any man; hell, he heals people simply by crushing certain aromatic leaves then laying his hands on you. He’s a descendant of Melian, and it’s clear that he’s got some residual power from that.
Finally, in the text, while Gandalf says the splinter was working inwards, he doesn’t say it’s headed toward the heart specifically. The heart is mentioned as related to the attempted stab of the Morgul blade; apparently if you are stabbed in the heart by such a magic blade, you don’t die, but become a wraith. It is, I think, a reasonable assumption that a splinter remaining in the wound and working deeper is likely seeking the heart, but it’s not specifically said.
And at the Ford of Bruinen, the Witchking is already almost all the way across the ford when the waters are released, and the sudden flood overwhelms him and two others. The rest are then driven into the water by the onrushing Glorfindel (in all his angry, other-world glory) and Aragorn and the other hobbits. Imagine a sudden flash flood without any warning: first surprise, then shock, then a reaction too late. Thankfully. 