There are multiple sets of rules, which conflict with each other in some points. Some parts of the rules rely on information not present in the spelling. And some words are pronounced irregularly, in a way that doesn't follow any of the sets of rules (or sometimes, different parts of a word are pronounced according to different sets of rules).
So as a whole, the pronunciation of Latin or Latin-looking words in English is not entirely predictable, although many specific parts of it are predictable (often highly so). But the fact is, if you speak English, you probably already have some intuition about the many reliable rules, so I'm not sure how helpful it is to formally study them. The parts that you are most likely to feel uncertain about are exactly the parts where the rules are least likely to be followed by other speakers.
Because of this, I would say the most practical way to get reliable information about the pronunciation of such words is not by learning and applying rules, but by using resources such as dictionaries (specialist ones, if necessary), or audio of specialists or experts in the field that uses the vocabulary that you are interested in. I wouldn't place too much trust in YouTube videos that are created to be pronunciation guides (as I think these are often automatically generated); I'd say Youglish is a better source, if you can find a clip where the word is pronounced in the context of real speech.
If you are interested in what kind of rules there are, here's a summary of the ones covering the areas mentioned in your question.
As Kate Bunting mentioned, there are 3 relatively common systems for pronouncing Latin words in general, which have been called things like "traditional English pronunciation", "Ecclesiastical pronunciation", and "restored pronunciation" respectively. (The names aren't very important; in fact, some of these names may be not very accurate descriptions.)
For botany, the "traditional English pronunciation" is by far the most important. This pronunciation follows very similar rules as the pronunciation of Latinate English words, such as the words various, traditional, combination, consistent. You can find details on this Wikipedia page: Traditional English pronunciation of Latin
The rule in this system about pronouncing vowel letters like the I in iris is as follows: when a vowel letter occurs in a stressed second-to-last syllable, and there is a single consonant letter (other than x) between it and the following vowel letter, the vowel letter is pronounced like its name/as a "long vowel"/like it would be when followed by silent e. So the ir- in Iris is pronounced like the ir in admire/admiring, more or less. (Accents differ in the extent to which a schwa-like sound may be present between the "long i" diphthong and the /r/ consonant sound, but the rules don't cover details like that.) However, some words have pronunciations that violate this rule by having a short vowel instead, such as syrinx.
The rule about the position of stress does not differ in most cases between the different systems. It is sometimes not possible to predict the regular position of the stress from the spelling of a word, because the rule is based partly on the word's etymology. It's regular in English to stress the syllable that was stressed in Latin, which had rules of stress based on syllable structure and vowel length. In a word like "virginica", the stress would regularly fall on the second-to-last syllable if the i in that syllable corresponded to a long vowel in Latin, and on the third-to-last syllable otherwise (if the vowel in the second-to-last syllable were short in Latin).
Virginica, it turns out, is a word that was formed after Latin was already dead. So even if we consider it a Latin word (which is arguable), you wouldn't find it in a dictionary of classical Latin. However, you can guess what the length of the vowel should be by considering that the ending appears to be the adjective-forming suffix -icus, -ica, -icum, (corresponding to the English suffix -ic), which had a short i sound in Latin. Therefore, the stress should regularly fall on the third-to-last syllable.