Does there exist an application or website that allows one to search for specific patterns with טעמי המקרא? For example, to find all occurrences of two consecutive זרקא's (as in וה' הכה כל בכור).
I am not aware of any such database, but you can create the sequences on your own, since you've mentioned that you are good in programming. Therefore I suggest you to download the text, and extract all teamim into a new file (or delete all other characters, which is acceptable according to most authorities). With regular expressions it's very easy. You may keep verse numbering to identify the places. To download the text, the following two sites are my suggestions:
- Mechon Mamre
By far the best text resource on the Internet, but it's copyrighted, and it's editing principles are a bit vague.
- Dovi's site on WikiSource
It may not be perfect, but everything is documented in Hebrew, so if you don't agree with a principle, you can change it.
It's the edition of the German Bible Society, which is one of the best scholarly editions. It's copyrighted again.
Most of these editions debate, whether one should follow more the Keter Aram Tzova or the Leningrad Codex. Another discussion, is to what extent should one accept the reconstruction of Mordechai Breuer (who advocated the Aleppo Codex).
Quantified Cantillation is what you need. It is designed to search the Torah for specific sequences of trop.
I can tell using it that there is one zarka-munach-zarka in Breishit 42:21 and one in Shmot 12:29, and a zarka-munach-munach-zarka in Vayikra 17:5.
- Torah only, not the rest of Tanach
- No searching backwards
It is open source, so you are welcome to add features.
Note: I have no affiliation with this program.