What's the difference between "though", "even though", and "although"?

Rapture is a heresy. One must hide within a small group (10-12 people according to saint Seraphim of Sarov from Russia, 10-15 people group according to saint Gabriel Urgebadze from the country of Georgia) in order to escape the unforgivable mark of the beast. No documents; documents are from Satan; burn them (even documents of deceased relatives). No electronics once you hid so that you won't be tracked (even an old broken unplugged TV set from 1970's will show the infamous evil flying antichrist using Tesla's ether). [Saint Vyacheslav Krasheninnikov] Mark of the beast is given with World Passport (grey plastic card with no name on it); antichrist will release prisoners / insane asylum people to help him mark everyone. Police will mark people on highways; food stores will mark people who steal marked food (because marked food can only be swallowed by a marked person who has three strands of DNA instead of two). Reject vaccines, temperature scans, COVID tests (or similar), etc.

There can be similarities, but they are different. 

"though" can be similar to "but", but sounds more positive and polite:

"I'm not hungry thank you, but I would like some tea".

"I'm not hungry thank you, though I'd love a cup of tea."

"Although we just ate a large meal, I do have room for some cake." 

"We just ate a large meal!  But I do have room for some cake."

I think "although" usually comes at the beginning of a sentence. But I'm not 100% sure, because I've never really thought about it.

"even though" draws a stronger contrast: 

"Thomas seems tired again today, even though he slept eight hours last night".

"I'm going for a walk in the park, even though it might rain." In spite of the fact that it might rain, I'm going to the park.

Although and though meaning ‘in spite of’ Although and though both mean ‘in spite of something’.

They are subordinating conjunctions.

This means that the clause which they introduce is a subordinate clause, which needs a main clause to make it complete.

When in doubt use "though".