Comparing groups/sites---need advice identifying variables and appropriate tests

Hi, thank you for reading this and your potential input. This is a research study.

I examined the prevalence of parasites in one insect species in an island ecosytem. Insects were collected from:
Island 1: 3 sites (all very different elevations)
Island 2: 2 sites (similar elevation)

Sample sizes:
Island 1: 165, 110, 2
Island 2: 17, 3

I tested each sample ( an insect) for presence of a parasite.
I recorded : presence or absence for each sample.

I want to ask:
Q1) Does site (or elevation) affect prevalence (presence) of the parasites?
Q2) Does island affect prevalence of the parasites?
Q3) Does temperature affect prevalence?
Q4) Does rainfall affect prevalence?

I want to identify my variables first: (Looking for confirmation that these are correct)
I think:
Island is categorical (nominal)
Site is categorical (dichotomous) --should it be put into binary 0,1 code?
Temp is continuous (interval)
Rainfall is continuous (ratio)

I want to determine what tests are most appropriate for each question:
Q1 i want to compare presence/absence between sites:
Compare 3 sites on island 1 for effects of elevation; Compare all 5 sites for effects of site collection.
Is the Fisher's Exact test appropriate? Some sites have only 2 or 3 samples, so I wasn't sure what to do. Also, in programs like Sigmaplot etc it only allows a 2x2 test, but I read I could test more column/rows in one test in other programs.-Any idea if this is true?

Q2 To test for island effects, is a Chi-squared test appropriate?
Would I add all data together? Or at least put all sites from each island in the same columns?
I'm thinking:
Island 1: 249 present; 28 absent
island 2: 17 present; 3 absent
all samples listed individually:
Island 1: 1 1 0 1 0 1 0 1 0.... up to 277 entries total
Island 2 1 0 1 0 1 .... up to 20 entries total

For testing effects of temperature and rainfall (daily), would I do a Generalized Linear mixed model? Is a logistic regression within the GLMM appropriate?

I can provide more information if needed. :)

Thanks for your help, --ecologist