Let's clear away the irrelevant complications, shall we?
The presenting question is equally clear with these examples:
Mary has the photo stuck on her wall.
Mary has the photo sticking on her wall.
Mary has the photo stick on her wall.
As noted, (1-2) contain past and present participles following and modifying the NP the photo, whereas (3) contains stick following the NP. Stick can't be present tense with a singular subject (sticks is singular), so it must be an infinitive without to.
These sentences are all grammatical, but a bit odd, since they mean different things and some of them require unusual contexts and connotations.
(1) is an informal way of saying that Mary's wall has the photo on it. The use of stuck (or in fact use of any form of the verb to stick) is unusual, because it refers to the adhesive coupling of the photo, instead of to the photo, or to Mary, or to the wall. This shifts attention to the method of adhesion, and stuck on her wall connotes a sloppy, possibly temporary job. Maybe it's a photo of her latest teen crush and will change soon. (1) is normal for some people, in some intimate contexts.
(2) goes even farther in focussing on adhesion, since it calls attention to the present stickiness of the photo, with the invited inference (since there must be some reason why the stickiness is relevant) that, while it's currently sticking on her wall, it may fall off in the near future. Adhesion is not normally an activity, and -ing participles are active.
(3) is a different construction from (1-2), since it could be any of several idiomatic have +
For instance, it could mean that Mary arranged for the photo to stick itself on the wall (perhaps via some mechanical arrangement). This would certainly be the meaning of the idiom if stick were transitive and its subject were an agent, e.g,
- Mary had Frank stick the photo on her wall.
But stick in (3) is intransitive, so it may be a different sense. Perhaps the photo is not very sticky and it's hard to mount, but Mary achieves success on her wall. She's lucky; she has the photo stick on her wall.
As I said, these are not very common situations, and mostly they are created by listeners who are trying to figure out why certain words have been used in certain ways. A lot depends on the imaginations of one's listeners.