(Note: Converting my comment to an answer after the edit)
There is a sense in which "carry" can point out someone who really does the main work (in this sense you might say that a sports player who is very good "carried his team to victory").
So in a conversation among multiple people--you might be pointing out certain individuals as the "they" who "carried a conversation". Without these people, it would have been silence otherwise. Perhaps everyone else would have "let it drop"--because they are bad conversationalists or didn't like each other enough to talk. :-)
He is disappointed that the women seem unable to carry a conversation.
What this might be saying is that he is disappointed that these women can't be good conversation leaders. They aren't able to ensure a good conversation goes on--no matter how bad the other people involved are at talking.
He is disappointed that the women seem unable to carry on a conversation.
Here they're not leaders, and moreover they're bad at participating at all. They aren't being asked to carry the whole conversation, merely to continue it a bit at a time...to carry on from the thing the last person said.
It's not a typical distinction--because we don't usually treat conversation as a competitive team sport, and talk about who "carried" it and who "dropped the ball". If you're not playing judge of how well people are doing at the conversation game, and just talking about what happened, then carry on is the right plain description for when "those people were talking".
Also: to single out someone and say they "carried on about something" would mean that they talked about it for a long time...generally too long. But when people "carry on" a conversation among themselves--without saying just one person "carried on"--it just means that the conversation kept going.
Why not just go with "has"?
This plan has many risks.
This plan has many rewards.
"Carries" has a sense of "brings with", e.g. the fleas carry a disease. So "the plan carries many risks" would be ok. But I would usually use "holds" for something concrete, e.g. the garage holds 20 trucks.
Best Answer
This a rather broad topic coming under the general description of collocations (Def. placement). There are verb-noun, verb-adverb,verb-verb, etc. They are idiomatic and there are no real rules, only guidelines in some cases. In this case, it should be "fulfill (BrE fulfil) a promise" and fulfill or meet a "commitment" (OED) http://oxforddictionary.so8848.com/search?word=commitment.
Some people will disagree on these, but most native speakers just know them intuitively: they sound right.
You can also look in Google Dictionary,
as well as: http://prowritingaid.com/en/Collocation/Dictionary?word=fulfil