Sure, there are. One interesting aspect of 'Bardock x Goku fanfiction' could be the exploration of their connection through the Saiyan bloodline. Since Bardock was Goku's father, some fanfictions might delve into how Goku might have been different if he had grown up with his father around. Maybe there could be stories where Bardock survives and trains Goku in the Saiyan ways from a young age, which would be a really cool and different take on the Dragon Ball story.
There is definitely some. However, the quality can vary greatly. Some might focus on their adventures together in a new setting, perhaps in a world where they have to team up against a new and powerful enemy. Others could be more about their relationship development, like how they first meet and gradually fall in love. You really have to dig through to find the ones that suit your taste.
A frequent theme in 'Bardock x Goku fanfiction' is power. How Bardock's power or knowledge could be transferred to Goku. Also, redemption can be a theme. Maybe Bardock has to make up for past mistakes through helping Goku. And of course, the theme of survival. Stories where Bardock somehow survives and his presence changes Goku's life in the Dragon Ball universe.
There are definitely some out there. However, the quality can vary a great deal. Some might focus on their friendship, while others might explore more of a bromance - type relationship. It really depends on what you're looking for in a fanfic.
Yes, there might be some out there. You can start by checking popular fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'bardock x king vegeta' on their search bars and you'll be presented with various fanfics related to this pairing.
At first, it might be tense. Bardock is a hardened Saiyan warrior from a different era, and Goku has grown up on Earth with a very different set of values. But as they start to communicate, they would find common ground in their Saiyan blood. Goku's pure - heartedness could gradually win over Bardock, and Bardock's strength and pride could earn Goku's respect.
I'm not sure of a specific highly recommended 'Bardock finds Goku' fanfic off the top of my head. But generally, fanfics with a high number of reviews and positive ratings are likely to be good. So when you search on FanFiction.net or AO3, sort the results by the number of reviews or ratings. Check out the ones that seem popular and read the summaries to see if they match what you're looking for in terms of Bardock finding Goku.
There might be some out there, but it can be a bit of a hunt. 'Lemon' fanfics often imply a more mature or risqué theme, which might not be allowed on all platforms. If you're really into the Goku and Caulifla pairing, you could also check out general fan communities where people might discuss or recommend fanfics that are more in line with the story and character development without going into inappropriate territory.
One popular 'bardock x goku fanfic' could be 'The Legacy of Bardock and Goku'. It often delves into the relationship between them in a unique way, exploring how Bardock's legacy might influence Goku's journey. Another could be 'Bardock's Impact on Goku' which might focus on the emotional and training aspects of their connection.
At first, it could be a bit rocky. Bardock is used to the harsh Saiyan ways, while Goku has been raised on Earth with different values. But as they spend time together, they start to understand and respect each other. Bardock could become a mentor to Goku, teaching him advanced Saiyan fighting techniques and Goku could teach Bardock about friendship and love.
One possible interaction is that Bardock could spar with Goku and Gohan. He'd test their limits and see how much they've learned about fighting. Another could be Bardock sharing stories of Planet Vegeta with them, which would be really interesting for Goku and Gohan to hear.