In many fanfictions, Natsu and Lucy's first kiss is often a very emotional and anticipated moment. Some writers might describe it as a spontaneous act during a dangerous adventure, like when they think they might not survive. Natsu, being his impulsive self, might suddenly realize his deep feelings for Lucy and plant a kiss on her lips.
It deepens their relationship immediately. After the first kiss, they become more aware of their feelings for each other. There's a new level of closeness and they start to rely on each other more emotionally. They might start to hold hands more often, and share more private conversations.
Yes. The kiss between Natsu and Lucy is a pivotal moment that adds depth to their relationship in the manga. It comes as a surprise to many but is a welcome development for their love story.
One common theme is the exploration of their unspoken feelings. Often, in the series, their relationship is somewhat ambiguous, and fanfiction allows for that to be explored more deeply with the kiss as a climax of those pent - up emotions.
Yes, there are quite a few. You can check out fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'Lucy and Natsu kiss' and you'll find a variety of stories written by different authors with different takes on their relationship.