Passwords: Two 50-characters vs one 100-characters

It depends on how they're verified.

If you provide both username and password, and the computer tells you "yes" or "no", then there are $62^{100}$ possible combinations: each of the $62^{50}$ possible usernames can be paired with each of the $62^{50}$ possible passwords to get $62^{50} * 62^{50} = 62^{100}$ possible username/password combinations.

If, on the other hand, you can independently check the username and password (for example, if the computer checks the username before checking the password), $62^{50} + 62^{50}$ is correct. You need $62^{50}$ guesses to figure out a valid username, and then $62^{50}$ more guesses to figure out the password.

You are correct, they are equivalent. The answer you quote adds $62^{50}$ and $62^{50}$ instead of multiplying these numbers together.

One hundred characters is one hundred characters. If it helps, think about what happens with smaller number of characters. (Take an extreme example, like only allowing the letters "a", "b", and "c" in the username and password, and comparing the number of one character usernames with one character passwords to the number of two character usernames.)

How would you write down a hundred character username? You'd write down a hundred characters. How would you write down a fifty character username together with a fifty character password? You'd write down a hundred characters. If you came accross a hundred characters somewhere, how would you know which process was used to create the string? You can translate any string of one hundred characters into either a username/password pair, or into just a username, and this translation is a bijection in each case.

Is this correct?

No. This is wrong: $62^{50} + 62^{50}$

There are $62^{50}$ possible user names, and for each one, there are $62^{50}$ possible passwords. Therefore, all together, there are $62^{50} \cdot 62^{50} = 62^{100}$ combinations.