Apostrophe doesn't make the previous consonant palatised, but soft sign does. Compare: з'є /zje/ vs. зье /zʲje/. Since the П /p/ can't be palatised in Ukrainian (/pʲ/ is impossible), apostrophe is the only option.
Apostrophe in Ukrainian (') is like the hard sign in Russian (ъ), compare Ukr. під'їзд and Russian подъезд.
This is a strange question. Why won't Russians use apostrophe? In fact, they did briefly in 1920-1930, here's an example from a 1922 book (Google Books, page 53, line 23):
In Church Slavonic writing, some letters could be written over line. When ъ (called єр back then) was written over the line, it got a simplified S-like shape called паєрок or єрок.
Here's an example from Wikipedia, рꙋ́сьскїи ꙗ҆зы́къ 'Ruthenian language' written with payerok in 1637 (here, ь and ъ are not distinguished; this happens in historical documents sometimes):
Russian Ъ evolved from a in-line yer, Ukrainian apostrophe ' evolved from over-line payerok/yerok.
It's impossible to say why Ukrainians choose to use over-line ' and not in-line Ъ. After all, why did English choose to use end-of-line s and dropped long ſ? Why not use ſ instead of s? It's impossible to answer such kinds of questions.
Ти is a familiar form, you use it to address the people you know well. Ви is formal or plural. To show respect, you speak to a person as if you're talking to a group of people; i.e. to show their opinion is as important as several people's, something like this.
When in doubt, Ви is almost always a safe choice. I usually use Ви to everyone and explicitly ask if I can start using ти. Ви might sound a bit too cold and formal, but it's safer to be formal than to be impolite.
When talking with Ukrainian speakers, you can simply ask them if they prefer ти or Ви. Actually, I do that quite often.
Ukrainian verbs have several forms, depending on how they are used in the sentence.
The main verb in the sentence changes its form to match the subject:
- я пʼю — I drink (1st person singular form),
- ти пʼєш — you drink (2nd person singular form; used in informal conversations),
- він пʼє — he drinks (3rd person singular form),
- вона́ пʼє — she drinks (—"—),
- воно́ пʼє — it drinks (—"—),
- солове́й пʼє — a/the nightingale drinks (—"—),
- ми пʼємо́ — we drink (1st person plural form),
- ви пʼєте́ — you drink (2nd person plural form; this form can be used when speaking to a single person for politeness),
- вони́ пʼють — they drink (3rd person plural form),
- соловʼї́ пʼю́ть — [the] nightingales drink (—"—).
We call those forms personal forms, because they change their form to show the person doing the action.
English also has something like this: it distinguishes between ‘drinks’ (3rd person singular form) and ‘drink’ (all the other form). Older English also had a separate 2nd person singular: thou drinkest. But Ukrainian has 6 forms.
Don’t worry if this seems overwhelming! You’ll learn this with practice. Those forms are not random, there are common patterns so you’ll be fine.
Пи́ти is the infinitive, it’s used when ‘to drink’ is not the main verb of the sentence. For example:
- я хо́чу пи́ти — I want to drink, I’m thirsty,
- ти хо́чеш пи́ти — you want to drink, you’re thirsty (informal form),
- вона́ хо́че пи́ти — she wants to drink, she is thirsty,
- ві́н хо́че пи́ти — he wants to drink, he’s thirsty,
- воно́ хо́че пи́ти — it wants to drink, it is thirsty,
- солове́й хо́че пи́ти — a/the nightingale wants to drink,
- ми хо́чемо пи́ти — we want to drink,
- ви хо́чете пи́ти — you want to drink (plural or polite form),
- вони́ хо́чуть пи́ти — they want to drink,
- соловʼї́ хо́чуть пи́ти — [the] nightingales want to drink.
Here, the verb ‘to want’ (хоті́ти) changes its form to хо́чу, хо́чеш, хо́че, etc. Пи́ти is not the main verb, so it doesn’t change its forms. Instead, a special impersonal form is used, called infinitive.