Does encrypting data multiple times using different passwords increase security? Does encrypting plain-text data multiple times using different passwords provide a greater level of security?

If so, what encryption algorithm would be best for such a use? Would it be beneficial to use different algorithms for each iteration?

I also had an idea to use 1 algorithm if the password starts with A-N and another if the password starts with O-Z. Is something like this beneficial?
There is a terminology gap in your question. Lets assume that you want to encrypt message $m$, and for that purpose you'll be using the encryption function $E$ such that the encrypted message will be $E(m)$. Now you ask whether your message will be more secure if you encrypt it for example as $E(E(E(m)))$. Well if this encryption is stronger than $E(m)$, then you might as well have selected in advance the algorithm $E'$ such that $E'(m)=E(E(E(m)))$. Does $E'(m)$ mean that you encrypt something 3 times? Well according to the custom terminology the answer is no, $E'(m)$ is a single iteration of the algorithm $E'$ which happens to consist of three iterations of $E$. If those three iterations were not to yield the optimal result then no one would bother to invent such algorithm $E'$. You may argue that the number of iterations represents some cost/performance tradeoff, but in this case the algorithm $E'$ would have been defined as $E'(m,n)=E^n(m)$ where $n$ is the number of iterations to employ (you'll find a similar concept in KDFs such as bscypt, scrypt, Argon2, etc. EDIT: But then a KDF is of course a one-way transformation as @tylo mentioned below, where multiple iterations do make sense). So the bottom line is that there is no such thing as "encrypting something multiple times", you'll always be just encrypting something, and this encryption may employ under the hood multiple iterations of one or more steps in the encrypting process.

Now to the point of your question, if you have multiple passwords then the strongest security you'll get is by concatenating these passwords together, pass them through a KDF, and use the resulting key for encryption. Your entropy for the entire process is the same, but if the encryption process is broken into iterations each one using only part of the entropy gives the attacker the luxury of keeping interim results and thus speeding up the cracking. So encrypting with a single iteration using a single password yields stronger encryption then multiple encryption iterations with multiple passwords which sum up to the same entropy as the password in the first option.

Concerning your idea of using different algorithms depending on password, well in that case you may leak information about your password if the attacker can guess the algorithm. So, you'll get better security by using an algorithm which is independent of message content or password.