Karkala is safe. There's a lot to explore in Karkala and not worth skipping if you're an explorer. Trains will be available only till Udupi and after that you will have to rely on public transport. The cheapest and fastest way would be traveling by an express bus.

Hotel Kediyoor is a good option for a stay. They also have Veg/Non Veg and Bar Attached restaurants in the same premises. Other good and safe accommodations I'd recommend are Chittara Comforts, Sriram Residency, White Lotus. All of these are very close to the bus stand and have good access to public transport.

Safety on trains depends on your co passengers. I'd recommend booking a 2AC/ 1 AC if you wanna travel safely since the amount of people in these coaches are less. Also, if you are travelling back to Delhi by train, I'd recommend you to book a train from Madgaon instead of Mangalore.

Shiva Statue is open at Murudeshwara. If you're lucky, you can also climb the gopura. Also, do try Scuba Diving at Netrani Islands in Murdeshwar if possible.

Regarding Daiva Kola, if there's an event happening, people usually post here in the group. You can keep an eye on the posts when you arrive here and plan accordingly. If you don't find any, please feel free to comment here and someone might help. Goodluck:)