Learn English – formal way of saying “I want to”

In formal writing you would just continually use "dear." See for example the correspondence between Vladimir Nabokov and Edmund Wilson, titled appropriately, Dear Bunny, Dear Volodya. In many (North American) business contexts, subsequent letters omit the greeting in favor of a more informal, chat-like style. However, "dear" is pretty neutral and almost always appropriate.

There are no settled greetings in informal correspondence. Much depends on the context and on the status of the person you are addressing (to state the obvious). Most importantly, the greetings and the sign-offs add a much-needed touch of personal flair. You can start a "hey" or a "greetings" or whatever else works for you. Pick your approach, and use it consistently.

As long as your tone is appropriate to the situation, you cannot go too wrong.

I am not sure if there are better alternatives but 'Ha ha!', 'Lolz', 'Rofl' etc., are the most common ways to show in an informal written dialogue or conversation that you are laughing without mentioning it explicitly. Of course, terms like 'Lolz' and 'Rofl' evolved thanks to the Internet usage but they have become quite a trend.

Your question, however, – 'how one can manage to assert that they are laughing without using euphemisms or colloquialism in first person', might be answered with a simple phrase like 'that makes me laugh' as Robusto suggests in the comments above.

Edit:

If this is what you are looking for (as I tried in my comment below) –

1) It's difficult not to laugh but the chicken wanted to cross the road because...
2) It's hard not to laugh but the chicken wanted to cross the road because...
3) Can't help laughing but the chicken wanted to cross the road because...