For example, if I have two copies of Battle Hymn in my hand and my opponent has a Propaganda, I could generate the two mana per creature needed to attack with all of my creatures. But if I cast them during Beginning of Combat, my understanding is that the mana drains before I can pay as I declare attackers. Is this correct, or am I missing something?
Not normally, since you can't cast spells in the declare attackers step until after the creatures have been declared as attackers, and the pool normally empties when moving from step to step. This means that the cost to attack must be paid before you can use the mana producing spells inside that step. However cards like Kruphix, God of Horizons, Upwelling or Omnath, Locus of Mana(green only) would prevent the pool from emptying, allowing you to use the mana in a later step.
No, you cannot use rituals or other instant speed effects that gain mana to pay for Propaganda or similar effects, and your reasoning for why this would be the case is correct.
First, as mentioned, rule 106.4 says
When an effect produces mana, that mana goes into a player’s mana pool. From there, it can be used to pay costs immediately, or it can stay in the player’s mana pool. Each player’s mana pool empties at the end of each step and phase.
So, any mana you produce in the Beginning of Combat step would empty from your mana pool before you declare attacks. Then, the Declare Attackers step is actually multiple sub-steps. The first step, described in rule 508.1, is to declare your attackers. This includes activating mana abilities and paying costs for declaring attackers (508.1i,j). After that, the active player gains priority, which would be the first time they have the opportunity to cast rituals or similar spells during the declare attackers step.
So, there is no opportunity between the mana pool emptying at the end of the Beginning of Combat step and paying to declare attackers during the Declare Attackers step to cast rituals.