I could hear him crying for help.

We would need all the help that we got. (Past tense)

I heard him (go/going) out.(What is the answer?)

Do they have any mistakes?
All these are OK:

I could hear him crying for help.
We would need all the help that we got. = We were going to need all the help that we got.

We needed all the help we could get.
I heard him go out.

I heard him going out.

Regarding the last two, be aware that 'going out' is an action that may take two or three minutes.

|..........the action of going out............|

|....................................................|< I heard this. I heard him go out. (the whole action)

|....................................................| < I heard this. I heard him going out. (some of the action)

CJ
Thank you, CJ!

Does the rule you mentioned regarding 'go' apply to 'cry'?
Teachers: We supply a list of EFL job vacancies
michaeltingDoes the rule you mentioned regarding 'go' apply to 'cry'?
Yes. It applies to all verbs in this construction:

I [heard / saw] [him / her / them / ...] [go / going / cry / crying / read a letter / reading a letter / ... ].

CJ