Part 1: Why option 4 is correct
I think the core of your question is 'why is option 4 the correct answer', because the clause 'examination seemingly long' looks incorrect to you, and you cannot see an error in option 3.
These are difficult sentences to parse, so I am not surprised you found them difficult.
In English, it is often possible to omit words which are simply repeating a grammatical element earlier in the sentence.
The examination seemingly long.
Is clearly wrong, because there is no verb. You need to add 'was'.
However, you can say:
The room was very warm and the examination seemingly long.
The second "was" is implied from the first part of the sentence.
The first part of the sentence also establishes a pattern:
noun, verb, adverb, adjective - the room was very warm.
The second part of the sentence needs to follow the same pattern, even with the omitted verb.
noun, [implicit verb], adverb, adjective - the examination [was] seemingly long.
If you said:
The room was very warm and the seemingly long examination.
you would create an incomplete sentence fragment. The 'seemingly long examination' no longer has an implicit 'was', because the examination and the room are no longer syntactically linked. 'Seemingly long' has become a adjective clause describing the examination, rather than the object of the clause 'the examination was ...' A reader would ask "the seemingly long examination was what?".
If it is still unclear, consider the following examples:
The cat was red.
The red cat was.
The first is a complete sentence. The second is a fragment where we do not know what the red cat was.
Part 2: Why option 3 is incorrect
There are two errors in option 3, although I imagine they are not easy to spot for someone learning the language.
Though it was very warm and the examination was seeming to be long, the students were trying to do the best.
The use of 'seem' and 'seem to be' is, I believe, rather idiomatic in English. See the dictionary definition of seem when used with adjectives (fourth section of the page). There is also a tense problem, but I'm unsure of the exact grammatical rules here myself, to be honest.
You could write the first part of the sentence as:
Though it was very warm and the examination seemed long, ...
Though it was very warm and the examination seemed to be long, ...
Though it was very warm and the examination was seeming long, ...
But not the version given to you in option 3. Or, you perhaps could say it, especially if it was part of a sentence on its own, but it sounds awkward.
The examination was seeming to be long.
is approximately acceptable English, but it simply sounds much better as
The examination seemed long.
Finally, 'to do one's best' is an English idiom, acting as a verb clause, and requires a possessive 'one's, my, your, their' to be included.
If you say 'to do the best', best is acting as an adjective, so you need to include the thing the adjective describes.
The students were trying to do the best [they could].
The students were trying to do the best [work possible in the heat].
"To do their best" says exactly the same thing, but does it by implication.