[RPG] lava-breathing lizard creature (that could be worshipped by a cult) in 5e

There are no rules for this, so it's always going to be up to the DM.

Polymorph can be used on "a creature you can see", with no further restrictions. It (for all intents and purposes) automatically fails against shapechangers and creatures with 0 hit points, but that's as far as it specifies. There's one clause in Polymorph which could be argued to apply here:

The target’s gear melds into the new form. The creature can’t activate, use, wield, or otherwise benefit from any of its equipment.

So if your DM considers a swallowed creature to be "gear" (unlikely), it would meld into the new form. Otherwise, Polymorph has no rules for this situation.

As you've noted, the rules within each creature's statblock have nothing that covers this situation either. It's noteworthy that, while each of them can only swallow creatures of a certain size, there's nothing that says what happens if a creature they've swallowed grows beyond that size. So, as a DM, I could certainly see an argument that when a creature you've swallowed becomes too big for you, the swallow ends messily.

Another point to consider about the rules for swallow within each creature's statblock are just that - within each creature's statblock. There are no global rules for swallow. Well, so, what?

The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast.

This means that, once Polymorph is cast, the rules that allowed the creature to have another creature inside it have ceased to apply to that creature. What does that mean? The only sensible way to deal with this paradox is to accept that the rules just don't cover this and the DM is going to have to decide what happens.

A few, but they all explicitly require DM's permission

As it is a method of communication, can it be taken as a language?

Not as a part of standart character generation. You will require an explicit DM's permission:

With your DM's permission, you can instead choose a language from the Exotic Languages table or a secret language, such as thieves’ cant or the tongue of druids.

Thieves' Cant is meant to be granted by a unique class feature. Even spells like Comprehend Languages do not allow you to understand Thieves' Cant.

Are there any other rule mechanics e.g. background, that could grant Thieves' Cant?

There are no "official" ways to get this feature other than taking a level in Rogue. However, with the DM's approval, you can get it from a custom background, homebrewed class archetype, or learn as a language.

Lore-wise, Thieves' Cant is anti-language or secret language is the jargon or language of a group, often employed to exclude or mislead people outside the group. It is a skill to hide secret messages in whatever language you are currently speaking.

Rules as written, a DM can allow Thieves' Cant as a custom background feature. See the PHB page 125 "Customizing a background":

If you can't find a feature that matches your desired background, work with your DM to create one

You can also learn Thieves' Cant during your downtime, if there's someone who agrees to teach you:

You can spend time between adventures learning a new language or training with a set of tools. Your DM might allow additional training options.

First, you must find an instructor willing to teach you. The DM determines how long it takes, and whether one or more ability checks are required.