Can a Player Character, say a 1st Elf Fighter, pick the Humanoid as their next Level, and become Elf/1st Fighter/1st Humanoid?
Also say you are a 20th Dwarf Cleric and you would rather not become Epic pick Humanoid and hence increase your BAB and Saves as per the Humanoid type instead of the Epic progression.
No.
There may well be ways for a player character to gain additional1 humanoid racial hit dice—none come to mind but I wouldn’t be surprised if there were a few—but just choosing “humanoid” as your level-up is not one of them. For that matter, a player character cannot choose to gain racial hit dice in that manner whatever their type—they cannot choose to level up in “dragon” or “outsider” even if they are a dragon or outsider.2
Player characters always gain a class level when they level up. Leveling up is described in the Player’s Handbook in quite some detail, and it always starts with choosing a class. There are some ways to increase your effective character level outside of the level-up process (templates), and even ways to get racial hit dice like that (lycanthropy, though that wouldn’t offer humanoid hit dice), but leveling up is always in a class.
For most purposes, commoner class levels are identical to humanoid racial hit dice. The most notable way in which they are different are that there are a few things (mostly templates) that scale based on racial hit dice, but not from class levels.
One way in which they are definitely not different is that they don’t change whether or not you are an epic character; these claims by the querent:
Also say you are a 20th Dwarf Cleric and you would rather not become Epic pick Humanoid and hence increase your BAB and Saves as per the Humanoid type instead of the Epic progression.
LA and Racial HD do not count towards your 20 character levels
are wrong. Your overall level is based on your effective character level, including racial hit dice and level adjustment. That is, in order to become an ECL 21st character, you need to level-up from 20th to 21st, and the XP threshold required to do that is a part of the epic rules. If you have RHD or LA, those will prevent you from getting 20 class levels without the epic rules. You can technically have both if you get the RHD and/or LA after you have the class levels (e.g. 20th-level character contracts lycanthropy), but even then, you still are an epic character because your ECL is above 20th.
Beyond any they start with due to being a member of a powerful race. Unusual for humanoids but a few such races exist.
There is one exception here, wherein a true dragon player character “is required to devote a level every few years to its dragon ‘class,’” per Draconomicon pg. 142. That is very much a special case and even then still refers to the process as a kinda-sorta “class.” This process is similar to the more codified racial classes and template classes in Savage Species and elsewhere, where a “class” is designed to gradually replicate the features of a powerful race or template, so a player character can choose that class for their level-ups and thus gain the race or template over time, including its racial hit dice or level adjustment. But racial classes and template classes are still classes first and foremost, to comply with the Player’s Handbook step 1 for leveling up: “Choose a class.” The Draconomicon dragon PC process is less codified and so may actually not be a true class but rather a “class” as Draconomicon puts it, and so is the only exception—that proves the rule.
Can a Player Character, say a 1st Elf Fighter, pick the Humanoid as their next Level, and become Elf/1st Fighter/1st Humanoid?
No, and there's no reason to. Racial Hit Dice tend to suck compared to real classes (the exception being Dragon which gives d12 HD, 3 good saves, and 6 skill points and I'd still rather have real class features). Humanoid gets d8 HD, a single save, and 2 skill points. A significant number of classes are strictly better than this (and the rest offer far, far more in exchange for slightly lower HD).
Racial Hit Dice are not classes- you do not choose to take levels in them. Monstrous races start with them. Savage Species kind of lets you take RHD but this is due to the its monster classes work- it slowly gains you that monster's statblock in a game starting at too low a level (its ECL is its RHD plus LA).
Also say you are a 20th Dwarf Cleric and you would rather not become Epic pick Humanoid and hence increase your BAB and Saves as per the Humanoid type instead of the Epic progression.
Even if this was an option, Racial Hit Dice count as levels for determining your character level. A Cleric 20/Humanoid 1 is still a level 21 character.
