Could we have a 0th dimension ? Could we have a -1th dimension ?
In the sense that a dimension number is a label, yes.
However we describe a space as having a given number of dimensions. In this sense a zero dimensional space would be a dimensionless nothing. A -1 dimensional space simply has no meaning in this sense and hence a -1 th dimension has no meaning.
Are we supporting the 2nd dimension, 1st dimensions existence by being part of the higher multiverse?
The dimensions are not related in that way. Again I'd suggest you let go of the view of individual dimensions and thing in terms of a space having a stated number of dimensions.
Finally, can fields inside our dimension (higgs field) interact with other dimensions? I am sure certain that we think gravity is a force interconnecting this "multiverse" together, if it even exists, but can our own particles and matter interact with dimensions completely different to us? Are we affecting other dimensions in this so called "multiverse"?
The dimensions, again, should not be thought of in this way. They are not things that interact with each other. I think you are mistakenly picking up this notion of interactions from the idea that space and time are linked in theoretical space-time models. But this "link" is really a description of the geometric properties of the space we use as our model and which incorporate time as a notional dimension. These are not interactions, but definitions of how that model of space is structured.
Even though this query makes absolutely no sense to me - I'm sure the second part of the question can be answered.
Note that an answer like "it makes no sense" would also be an answer. I think the problem here is that you are treating dimensions as things.
The "right" number of dimensions is a question that starts heated arguments among physicists. If better minds than mine can't agree I'd be loath to express an opinion. I'll go with whatever works for whatever system I need to model, and for an awful lot of purposes Newtonian still works fine.