Learn English – Should I use “at the end of” or “in the end of” in this sentence

With the future, the infinitive form expresses intention whereas will + <VERB> can express either an intention or the inevitable.

Consider this scenario:

Thousands of scientists worldwide agree that the polar ice cap will melt, based on their careful observations, unless measures are taken to dramatically reduce emissions of so-called greenhouse gases.

How might the headline read?

POLAR ICE-CAP TO MELT

POLAR ICE-CAP WILL MELT

Does the polar-ice cap intend to melt? No. So the verb would be will.

But let's say a government plans to cut the budget of its environmental watchdog agency. That is their intention.

GOVT TO CUT BUDGET OF ENVIRONMENTAL AGENCY

GOVT WILL CUT BUDGET OF ENVIRONMENTAL AGENCY

There, the choice is less clear. Probably the infinitive, but not necessarily.

By preposition Indicating a deadline or the end of a particular time period. ‘I've got to do this report by Monday’ - ODO

Until preposition and conjunction Up to (the point in time or the event mentioned)
(as preposition) ‘the kidnappers have given us until October 11th to deliver the documents’
(as conjunction) ‘you don't know what you can achieve until you try’
- ODO

Common Mistakes in Business English notes:

• Use “by” when you refer to a deadline.
• Use “until” when you refer to the period of time before a deadline.

English With a Twist notes:

Until – describes a period of time before the deadline. In other words, up to a particular time.
By – is a time preposition and means “on” or “before”. It tells you when.

The key difference is that by focuses on the deadline while until focuses on the period before the deadline.

Here are your quotes, which I'll number for convenience:

1. I've got to pay the money back by the end of the month.
2. I've got until the end of the month to pay the money back.
3. I put on an extra pair of socks. Until/By then me feet were freezing cold.
4. I stood outside the cinema for an hour. By then my feet were freezing cold.

Example 1 focuses on the deadline itself (date of final payment), whereas example 2 focuses on the period prior to the deadline (period when payments are accepted).

Examples 3 and 4 with by speak of the time when the feet could be observed to be "freezing cold", whereas with until, they speak of the period of time when the feet were "freezing cold". There is a measure of intent that has to be inferred in these examples, about whether the end-point or the duration was the focus; it's not completely precise one way or the other.

The context of #3 suggests that the "freezing cold" applied prior to wearing the extra socks. The focus of the "freezing cold" sentence could be on the period prior to wearing socks (use until), or it could be on the time at which the socks were put on (use by).

On the other hand, the context of #4 suggests that the focus is on the time at the conclusion of the hour, so by is more appropriate. Although it talks about a time period, one's feet normally don't start off "freezing cold", hence the "freezing cold" reference applies to the end of the hour, not to the whole hour. Had the context sentence been that the person stopped standing outside after an hour, the inference would be stronger that they had concluded their period of having cold feet - the period then making until more appropriate.