A frequent theme is the idea of two heroes finding love and support in each other. Given their similar goals of protecting and doing what's right, it makes sense that in fanfic they would find a deeper connection. Also, the theme of overcoming differences, not just cultural but also in their powers and leadership styles, is quite common.
Yes, there are some out there. You can start by looking on fanfiction platforms like Archive of Our Own. Some stories might explore their friendship in a different, more romantic light, with interesting plotlines and character development.
Yes, there are some. In the fanfic community, there are various interpretations of relationships between characters. However, these are mostly works of fan creativity and not part of the official Marvel canon.
Common themes often include them teaming up in unique ways that lead to a deeper emotional connection. For example, they might go on a mission together that forces them to rely on each other more than usual, and this could be the start of a relationship in the fanfic. Also, themes of mutual respect and admiration are frequently explored, as they are both heroes with different but equally valuable skills and qualities. Another aspect could be the exploration of how they deal with the public's perception of their relationship if it was more than just a friendship in the fanfic world.
Archive of Our Own (AO3) is a great place to start. It has a vast collection of fanfics, and you can easily search for 'Gay Captain America' to find relevant stories. The tagging system there is very helpful for narrowing down your search.
Captain America has indeed yielded Thor's hammer in the comics. It happened when his character's integrity and moral compass proved him worthy of handling such a powerful weapon.
Captain America has lifted Thor's hammer in the comics. It's often a moment of great significance, demonstrating his pure heart and heroic nature. These scenes add depth to the characters and the overall narrative of the comics.