Yes, as your source seems to indicate as well, if on its own can have the meaning of even if.

In the given examples, even is omitted but the meaning is still there.

I will beat this game if it's the last thing I'll ever do!

is equivalent to

I will beat this game even if it's the last thing I'll ever do!

No, there are cases where if you omit the "even", it will sound strange. Those sentences you gave as examples are extremes (and also set phrases, almost idioms), so people will understand that you omitted the "even" from context. But other less extreme/familiar sentences would be less clear, and therefore be strange.

For example, "Take off your sweater if you are cold." does not make sense (why would someone tell you to take off a sweater when you are cold?), but "Take off your sweater even if you are cold" makes sense because now the speaker is saying to take off the sweater despite the fact that you are cold.

I also think leaving out the "even" is sort of slangish and considered "incorrect" in more formal English, though people will understand it in those particular examples. It's similar to how people say, "I could care less," when they mean "I could not care less," yet people understand their meaning due to the fact that the phrase is used often, and from context.

The word "even" in "even if" is implied in cases where the condition is particularly strong. Omitting it in these cases would not alter the meaning of the sentence and one can argue that it is in fact redundant as the strong condition already provides emphasis.

In the following case:

I wouldn't marry you if you were the last man in the world.

"last man in the world" is an extremely strong condition and therefore provides emphasis on its own. Adding "even" does not change the meaning of the sentence and is not necessary here.