It kind of does. Correctly using "עכשיו" alongside a verb can only mean it's happening at this very instance. If you want to rather say "the pigeon will come soon", for example, it should be "היונה תבוא בקרוב/עוד מעט"; if you want to refer to something that occurs regularly in present time, you should use "כיום" (another way of saying "these days" ["בימים אלו"]), which doesn't necessarily indicate this exact moment.
Yes, it is vav. In ancient Hebrew it may have been pronounced like W, however in modern Hebrew it is pronounced as a V sound, and sometimes vav represents the vowels U and O. So, they are V, O, U :)
אני ואתה, את ואני, אני ואת
Ani ve'ata, at ve'ani, ani ve'at (and many others:)), here vav means "and" and sounds like "ve"
That's actually not entirely correct, as Vav HaChibur has different pronunciations depending on the word it's preceding.
Most Hebrew speakers are not entirely consistent with this, but, for example:
When the word begins with the letters bet/vav/mem/peh (ב/ו/מ/פ) or with any letter other than yod (י) having the shva nikud (אְ), the vav is actually read as "woo". For example: בָּנִים וּבָנוֹת (banim woo'banot); אַחַת וּשְׁתַּיִם (achat woo'shtayim).
When the word begins with a yod having the shva nikud, the vav takes a hirik nikud (וִ). For example, יְהוּדָה וִירוּשָׁלַיִם (yehuda vi'irushalayim).
When the word begins with a chataf nikud (אֲ/אֳ/אֱ), the vav takes the nikud of that letter (though written without the chataf). For example: אַתְּ וַאֲנִי (at va'ani). Things get even weirder for chataf kamatz (אֳ) because it's pronounced as "o" (small kamatz) while a regular kamatz preceding it is commonly read as "o" as well (צָהֳרַיִם – tsohorayim, though the Sephardic pronunciation maintains the first kamatz is a big kamatz, so they pronounce it as "tsahorayim"). So if a word begins with a chataf kamatz (I can't find any common ones, though), perhaps the overly-correct way to pronounce a vav preceding it is as "vo-".
When the vav is part of a common phrase, that vav takes a patach nikud (וַ). For example: בָּשָׂר וָדָם (basar va'dam – flesh and blood); יוֹמָם וָלַיְלָה (yomam va'layla – day and night).
When preceding the words "reva" (quarter) and "hetzi" (half), it will also take a patach nikud: אַחַת וָרֶבַע (achat va'reva); שְׁתַּיִם וָחֵצִי (shtayim va'hetzi).
Mind you that most Hebrew speakers aren't really aware of all of these detailed rules and may or may not speak in compliance to them. I didn't remember all of this and had to find an online source in order to provide this answer.
The order is usually "היונה באה" (the pigeon came/is coming), but you sometimes hear the reverse order in more "decorative" speech (I'm not sure if I would say it's necessarily used in formal speech, but there are, for example, Israeli songs that use verb–noun). I would say it's more common in question form, but it's still more common to hear noun–verb questions.
I'm just starting Hebrew, and I feel like I'm not getting enough info from the app. Is there a better place to understand the basic constructions of Hebrew? It seems like letters make different sounds based on context, and I can't follow why. Like, sometimes a letter will make a certain consonant followed by a certain vowel sound, and other times it seems like the consonant sounds very different and the vowel is a different vowel together. I know learning the writing system is important, but I do kind of wish there was an option to show the Romanization (I hope that what you call it? That's what it is for Japanese) underneath the real way it's written, at least until I can consistently understand how letters are meant to be pronounced.
In general the consonants in Hebrew (which is partially an abjad) are consistent, but vowels are only indicated using letters (and not always) if the sound is "-ee" using the letter yod/י (for example, a Person [male/female] is "איש/אישה" "ish/isha") and if the sound is "-o" or "-oo" using the letter vav/ו (for example, Skin is "עור" "or", Body is "גוף" "goof"). In the middle of words the vowels "-ah" and "-eh" are rarely indicated using letters; "-ah" might be indicated in the middle of words using alef/א in some rare occasions or in some transliterations from foreign languages. At the end of words the letter heh/ה is used for either vowel, and sometimes the letter alef/א for "-ah" (also often foreign transliterations or words that entered Hebrew from Aramaic, but not always) or "-eh" in a few words. The letter heh/ה can also seldom indicate the vowel "-o" at the end of a word (Shlomo [Solomon] is "שלמה"), and the letter alef/א in some situations indicates an "-o" (Head is "ראש" "rosh") or "-ee" vowel (First or Sunday is "ראשון" "rishon"). An example of alef/א also appearing in the middle of a word while indicating an "-ah" vowel is the word Neck "צוואר" "tzavar".
As you can see, it's quite complex and basically the correct pronunciation of different words eventually comes from familiarity with them. Sometimes you may even have several different ways to pronounce a certain series of letters (mostly related but different words written the same way) and you have to infer from the context of the sentence what would be the correct pronunciation.
There has been in Hebrew since the Middle Ages a system of diactritics called Nikud, which isn't usually used in texts written for adults (other than religious texts or when clarifying a specific word) but is used when teaching Hebrew at elementary school.
Duolingo doesn't seem to feature them, but perhaps it would be a good suggestion to include it (along with explanations on how to decipher the Nikud signs, as it could be tricky at times as well).
Back to consonants, you have some related variations you may encounter, but they mostly have rules that can help indicate what is the correct version of the consonant.
You have the Dagesh sign (a dot within a letter) which in modern days can appear (but is usually visually omitted if no Nikud is used) in the letters bet/בּ, kaf/כּ and peh/פּ (historically it also existed for the letters gimel/ג dalet/ד and tav/ת but today it doesn't affect their pronunciation). When these letters have a Dagesh, they are analogous to B, K and P, respectively. When they don't have a Dagesh (meaning the letter is Rafah, soft), they are analogous to V, Kh (similar to the J in "jalapeño") and F, respectively.
You also have Shin and Sin ("right shin" and "left shin", שׁ and שׂ [notice the diacritic]) for either "sh" or "s" sound.
The Shin/Sin variation purely depends on the specific word, but the Dagesh pronunciation has some rules. The rules are a bit complex and most people don't really know them all completely, but the basic indication is that the Dagesh is found when the respective letter is the first one in the word or the first after definite article Heh Hayediah/ה הידיעה (which could be simply "the" "הַ/ha" or "to the" "לַ/la"), for example:
A dog is "כֶּלֶב/kelev", the dog is "הַכֶּלֶב/hakelev", to the dog is "לַכֶּלֶב/lakelev", but TO A dog is "לְכֶלֶב/lekhelev" (like the J in jalapeño).
There are also rules that involve what type of Nikud was on the previous letter, and a couple of other stuff, but they are too complex to explain here and either way most people either talk correctly due to familiarity with each word or just talk incorrectly (sometimes the very most correct way of pronouncing a word or a word within a certain context might sound archaic).
Hope that helps a bit.