На вечеринке весело.

Because after на when it means at/on a location, Russian uses the prepositional case.

• dictionary form: вечеринка = party
• after на for location: на вечеринке = at the party

So the ending changes from -а to -е.

Russian usually uses на with many events and activities:

• на вечеринке = at the party
• на концерте = at the concert
• на уроке = in class / at the lesson
• на работе = at work

Even though English often uses at, Russian often uses на for this kind of situation. It is just the normal idiomatic choice.

In the present tense, Russian usually leaves out to be.

So:

• На вечеринке весело.

literally works like:

• At the party, [it is] fun.

You do not need a word like is here. That is completely normal in Russian.

Here весело is best understood as a predicative adverb or state word. It describes the situation as a whole: it is fun / it is lively / it is cheerful.

It is not agreeing with вечеринке.
It is not saying the party is a веселая thing in the grammatical sense. Instead, it means that the atmosphere there is fun.

So the sentence means something like:

• It’s fun at the party
• The party is lively
• There’s a fun atmosphere at the party

depending on context.

Not really. This is an impersonal sentence.

English often uses a dummy subject like it:

• It is fun at the party

Russian does not need that dummy subject here. So весело can stand by itself as the predicate of an impersonal sentence.

Good question. Russian ё is always stressed. When the stress moves away, it often changes to е in related forms.

Compare:

• весёлый = cheerful, fun
• весело = fun, cheerful, merrily

In весёлый, the stress is on ё.
In весело, the stress moves to the first syllable: вЕсело. Because ё cannot be unstressed, you get е instead.

So this change is normal.

They are close in meaning, but not identical.

• На вечеринке весело = It’s fun at the party
  This describes the atmosphere or situation.

• Вечеринка весёлая = The party is fun / cheerful
  This directly describes the noun вечеринка with an adjective.

The first version is very natural when talking about what it feels like there.
The second is also possible, but it sounds more like you are characterizing the party itself.

Yes. Russian word order is flexible.

• На вечеринке весело = neutral, natural
• Весело на вечеринке = more emphasis on fun
• На вечеринке-то весело = stronger contrast or emphasis in conversation

The basic meaning stays similar, but the focus changes. The version you were given is a very normal neutral order.

Yes. That changes the meaning slightly.

• На вечеринке весело = It’s fun at the party / The atmosphere is fun there
• Мне весело на вечеринке = I’m having fun at the party

Adding мне makes it specifically about your experience. Without мне, it is a general statement about the party atmosphere.

Not exactly. Весело usually means something like:

• fun
• cheerful
• lively
• merry

It often describes a lively, enjoyable atmosphere, not just an internal feeling of happiness.

So in this sentence, весело is closer to it’s fun or it’s lively than to someone is happy.