How can i know when to use -ing or -ed or infinitive without to?
We will soon have your car "going". (Why use "going"?)
It's nice to have people "smile" at you in the street (why use "smile"?)
Concerning your sentence with smile, I'll quote from one PDF by Marcus Callies that quotes Biber and Huddleston&Pullum:
verbs taking the bare inf. come from just two semantic classes:
(according to Biber:)
• perception verbs (feel, hear (tell), see, watch);
• verbs of modality or causation (dare, have, help, let, make)
(according to Huddleston&Pullum:)
sensory perception verbs (feel, hear, notice, observe, overhear, see, watch) • have, let and make among the causatives
• a few are found either with or without to: ought, dare, know and find (typical of BrE), help
In your second sentence, have is a verb of causation: you do something that causes people to smile at you. You can use either smile or smiling, with have but not with make, for example.
Concerning the car going, I guess we can use go but the meaning will change to "we will launch your car" or "we will make your car go away".
You cannot use the -ed form in either of your sentences, it would be ungrammatical
We will soon have your car went. (WRONG)
It's nice to have people smiled at you in the street. (WRONG)
I'm not 100% sure why, but it's probably because smiled and went are finite verbs, and in some constructions you are only allowed to use non-finite verbs. It's probably a "catenative construction" in your case: one verb connects to another verb.