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.