I don't understand why the ING form is used in these examples

The simple and continuous forms of all verbs have their respective nuances and you should decide what you want the sentence to mean, and then choose the appropriate form.

The continuous form of the verb indicates

(i) an action that is/was/will be (i) incomplete and (ii) in progress (iii) at the time that is being referred to (it has started but it has not yet finished) -> I will be/am/was/have been/had been riding a bike = I will be/am/was/have been/had been in the process of riding a bike but have not yet finished riding the bike at the time I am referring to.

The continuous form used to be known as “the imperfect”: It was called “imperfect” because the action had not been “perfected” i.e. it had not finished.

OED 5. Grammar. Applied to a tense which denotes action going on but not completed; usually to the past tense of incomplete or progressive action.

1871 H. J. Roby Gram. Latin Lang. §549 Three [tenses] denoting incomplete action; the Present, Future, and Imperfect (sometimes called respectively, present imperfect, future imperfect, past imperfect).

All simple forms of the verb indicate an action as a whole - from start to finish.

The simple form of the verb can indicate a habitual or regular action that

(i) is/was/will be complete/completed each time it is undertaken.

A: What do you do to keep fit?

B: I ride a bike. -> “ride” includes everything from getting on the bike at the start of the journey to getting off the bike at the end.

Or

(ii) a single, complete or completed present, future, or past action:

"He told me that I had to visit the Eiffel Tower, so I go/went/will go to Paris on Wednesday” -> “go/went/will go” includes everything from the decision being made, bags being packed, going to the airport, etc., to the arrival in Paris.

(iii) a habitual, recurring, regular or frequent action (that is completed each time)

On Saturdays, I go to the gym.

He ate toast for breakfast every day of his life.

This is my last night working here.

This is similar to a relative clause:

This is the last night that I am working here.

The subject of working is left understood and cannot be inserted. Similar gerund-participials are permitted by all sorts of words denoting spans of time: hour, minute, day, year, etc. as long as there is a determiner supplying the understood subject.

There is a problem sending this email.

In 2, sending this email is similar to a relative construction as well:

There is a problem that occurs when sending this email

compare with:

There is a scammer sending this email.

However, we understand that the problem is not doing the sending and instead take sending this email as telling us when the problem occurs.

In 3, facing him could be understood as an adjunct of reason or condition

I doubt the chances of success facing him.

Since we are facing him, I doubt the chances of success.

If we are facing him, I doubt the chances of success.

Or, like the others, it could be understood like a relative

I doubt the chances of success that are facing him

For all three constructions, there is no one clear interpretation dictated by the grammar, but there is a likely one that most people reading them would get.

In 1 and 2, my last night and a problem are not likely candidates for the subjects of working here and sending this email. This makes an interpretation similar to that of relative clauses likely.

In 3, something similar is happening in that chances are not usually said to face people. This pushes us towards interpreting facing him as the reason for doubting and guessing at the subject of facing him from the context.

As far as the reason for choosing gerund-participials here and not to-ifinitivals, to-infinitivals would force a different interpretation:

my last night to work here

= the last night I have a chance to work here

a problem to send this email

=a problem (for someone) to send this email

to face him

= (purposive) to face him / in order to face him

= which will face him (at some future time)