Is it correct that dependent origination will cease to exist if one doesn't have 12 nidanas?

I can only try to answer your first question like this:

Dependent origination is - among several things - a theory to describe how our perceptions are created. As such it's not a question whether you have the nidanas or not. To make a comparison, that would be similar to asking whether we can have gravity or not. Gravity is present under certain circumstances, and is absent under other circumstances.

Dependent origination can perhaps be compared to the western idea of an ontological axiom, meaning it's a theoretical framework to interpret and understanding how phenomena arise and cease.

There are five Niyamas subject to Mode of Conditioning applied for mental and physical. The way I understand When Buddha taught Dependent Origination he was talking about the arising of Dukkha hence the application of mental phenomena. Hence Arhant Body (Physical) is still subject to DO. However, his mental factors subject to DO has ceased. (ending birth)

http://103.242.110.22/theravadins/English-articles/abhidhamma-in-practice.pdf

will dependent origination cease to exist if one doesn't have 12 nidanas?

In Buddhism, what should be uprooted is Ignorance (Avidyā, Avijja) which is the first nidana in dependent origin. This results to cease the other 11 nidanas in dependent origin.

And in general, how does one know that all dependent originations of a thing are "depleted"?

When someone uprooted the Ignorance it becomes very clear to him. Because who ever in the path to Nivana knows what's ignorance and when they uprooted they just know there's no more Ignorance in the mind.

will dependent origination cease to exist if one doesn't have 12 nidanas?

Yes.

In my understanding, dependant origination is just an extended understanding of "condition". For example, if I say "plants need water, soil and light to grow", then water, soil and light are the originations. This doesn't seem to relate to the mental condition of the observer. So how do the two relate?

'Paticcasumuppada' ('Dependent Co-Arising') is the subset of 'Idappaccayatā' ('Conditionality') that explains the 12 conditions for the arising of mental suffering. The suttas say:

And what is dependent co-arising? From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging. From clinging as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.

The conditionality related to the arising or coming-to-be of trees is not called 'Paticcasumuppada'. The arising of trees is merely an example of general 'Idappaccayatā'.

And in general, how does one know that all dependent originations of a thing are "depleted"?

This question is not valid because not all examples or occurrences of 'Idappaccayatā' are 'Paticcasumuppada'. The mind (while living) can experience the depletion of 'Paticcasumuppada' but the mind (while living) cannot experience the depletion of 'Idappaccayatā'. Even though the suffering of 'Paticcasumuppada' can end; the 'Idappaccayatā' of Nature will continue to be discerned. In short, suffering can end but trees will continue to grow, clouds will continue to form, the body will continue to live nourished by food, consciousness will continue to operate conditioned by sense organs, etc.