Wow, Sagitta you are so wrong on your English lately! It's programmer in American English. The reason for the double M is that keeps the A short. Without a double M, the E makes the A a long A. Like "gamer" versus "spammer" or "cam/cammer" versus "came," "run/runner" versus "rune"
You can find anything outside of the standard on the Internet. I have never seen it "programer" and it looks incredibly incorrect this way, not to mention the pronunciation for it looks like /proʊ-'ɡrɛɪ-mər/ instead of the correct /'proʊ-ˌɡræ-mər/. The point is, English spelling rules dictate that vowels E and I change the previous vowel in a word to its long sound. A double consonant nullifies the E or I effect and keeps the previous vowel's short sound.
duke - duking duck - ducking
Hey, I agree with you, I agree that the pronunciation should be different in this case and strange like that, I haven't seen it anywhere either, but it's in the dictionaries, and there are native speakers confirming that they want it to be accepted :')
(if it wasn't in the dictionaries, or if they weren't asking, we wouldn't have had it here...)
The ґ here is wrong
In this course "є" (in this kind of sentences) is not accepted because
1) It is not natural in standard Ukrainian
Using "є" is obsolete in modern Ukrainian and sounds unnatural. It is pretty common in some regions of Ukraine though, but not in the "textbook" Ukrainian.
2) Copying the structure of the English sentence
The idea is, if a learner actually does not speak any Ukrainian, they might write "є" simply because they are copying the structure of the English sentence. The developers of this course think that the users should learn that it's unnatural, even though it's technically not wrong grammatically. And if the users do actually speak some Ukrainian with the variation/dialect where "є" is natural (e.g. some Western Ukraine regions, Canadian born Ukrainians), they can understand that this course is also for people who don't speak Ukrainian at all and just go along with it, speaking the way they want at home, and following the "textbook rules" here :)
The letter Ґ is definitely wrong. It is very rarely used in Ukrainian.
In past and future tenses, the object does take the instrumental case.
Мій син був програмістом. Мій син буде програмістом.
Ukrainian has a lot of influence from Polish as well, and Polish does not omit the present tense verb. Mój syn jest programistą, and the object takes the instrumental case.