비행기를 보는 사람이 많습니다. [pihaenggi-reul po-neun saram-i manh-seumni-da]
Presumably, 이 is the verb "to be"
Here, 이 is the subject marker. 많습니다 on its own conveys the full meaning of 'there are a lot', so there's no need for another verb 'to be'.
착륙하는 비행기가 많습니까? [ch'angnyuk'ha-neun pihaenggi-ga manh-seumni-kka]
with the verb "to be" omitted ...
First, the choice of -다 as "base dictionary form" is, in a sense, an arbitrary convention.
Unlike English, Korean verbs always require a suffix. You will never see someone just saying "많", "없", or "아름답". It's simply not grammatical at all - that's like someone saying "Engl" instead of English. Because of ...
You can say:
카페도 많고 식당도 많습니다.
카페와 식당이 많습니다. (These two are almost equivalent, I personally can't tell any difference.)
카페도 식당도 많습니다. (This also makes sense, but my personal impression is that this form sounds more natural when used with negative adjectives. For example: 카페도 식당도 없습니다. You can also add 다, like 돈도 명예도 사랑도 다 싫다.)
Something not relevant but I ...
'을' is an object particle(목적격/대격 조사). In that case, '펜' should be an object, and we need a transitive verb. In the sentence '있다' seems to have the meaning 'to have', but lexically it just means 'to exist' or 'to be'. Thus '있다' does not require an object. the correct sentence should be either 저는 펜을 가지고 있어요 or 저는 펜이 있어요
기 at the end of a verb creates a noun form of the verb, like the gerund ('-ing') in English.
팬(을) 만난다 = I meet fans. (-ㄴ다: present tense sentence ending form)
팬(을) 만나기 = meeting fans. (-기: suffix creating a noun form)
(subject/object particles are usually not included in this type of phrasing)
There are altogether three ways to create a noun form.
Are there any specific rules for these particles—
These kinds of particles have two variants,
one for those with a final consonant (e.g. “잭” — “은,” “이,” “을,” “과,” “-이다,” “-이면,” “-이라서,” “-이라면,” “-이고,” “-이었-,” “-이에요,” and “-아.”),
and another for those without (e.g. “사과” — “는,” “가,” “를,” “와,” “-다,” “-면,” “-라서,” “-라면,” “-고,” “-였-,” “-예요,” and “-야.”).
does “지안은 철수를 좋아한다” mean that 지안, not someone else, likes 철수?
By itself, no, but in a certain context it could. E.g. 민지는 지민을 좋아하는데 지안은 철수를 좋아한다.
Or does it mean that 지안 likes, not hates, 철수?
No, nothing about that sentence in itself implies this nuance. This kind of implication would be done through intonation when speaking.
If person A asked who broke ...
"오늘은 일찍 왔어요" is similar to "Today I came early."
You probably wouldn't say that if you come early every day: the sentence implies that it's something worth talking about. At the same time, it doesn't emphasize anything being unusual. (For example, maybe today I came early because it's Tuesday and I come early every Tuesday.)
-을 after time duration usually gives a sense that this time duration is a fresh topic that is being brought up. I don't consider this -을 to be object marker.
For instance, if someone asked you about how long you've been living in Korea, then you would usually say: 2년 살았어 or 2년동안 살았어. It's without -을 with time duration because the duration was the topic of ...