Can "that" as a pronoun be used without a leading noun?

I

The pronoun to use in this context is normally "it". As explained in this article, "Whether the “getting pointed out/at” feature is particularly relevant depends on the specific intent of the speaker, and isn’t always necessary or meaningful.". The "getting pointed at" is not necessary here, as to the meaningfulness you must make certain, for instance, that your context makes for a singling out of this object among others.

Given your specification, the sentence is not correct because you have two clauses side by side without punctuation or linking word to articulate them. At least some connecting punctuation is needed in this case. This connection, so called by parataxis, gives of course the option of using a full stop, which doen't offer perhaps the most interesting means of writing those two juxtaposed ideas.

• An apple is on the table. I looked at that.

A simple comma will be more usual maybe, but as well a semi-colon will do.

• An apple is on the table, I looked at that.

Otherwise, some other means of connection must be used. The simplest is the use of the coordinating conjunction "and". However, without changing the tense of the first verb this could result in an incoherent coordination.

• An apple is on the table and I looked at that.
• An apple was on the table and I looked at that.

Another possibility of connection is to make the second idea dependent on the first by using a subordinating conjunction; the relation of causality is not very definite but yet it can be argued it exists.

• An apple is/was on the table so I looked at that.

II This second sentence is correct but it seems that it would be put differently: "What he likes is good.".

III The style of this third sentence is very informal. The plain reading of this makes the professors into "a situation". Anyway, the first "that" is not a demonstrative in this case; the second is a demonstrative pronoun and there is no problem.

• While some might think that all professors do is lecture a few times a week and assign some homework, that's not true for all profs.

In the second example the first "that" is a conjunction (or a conjunctive participle.) The OED explains how it is a conjunction:

that, conj.

I. 1.a. Introducing a subordinate noun clause, as subject, object, or other element of the principal clause, or as complement of a n. or adj., or in apposition to a n. therein. The subordinate clause as subject is most commonly placed after the verb and introduced by a preceding it, e.g. ‘it is certain that he was there’ = ‘that he was there, is certain’. As object, it usually follows, e.g. ‘I have heard that he was there’.

[This use of that is generally held to have arisen out of the demonstrative pronoun pointing to the clause which it introduces. Compare

(1) He once lived here: we all know thát;

(2) That (now this) we all know: he once lived here;

(3) We all know that (or this): he once lived here;

(4) We all know thăt he once lived here;

(5) We all know he once lived here.

In senses 1 – 3 that is a demonstrative pronoun in apposition to the statement ‘he once lived here’; in sense 4 it has sunk into a conjunctive particle, and (like the relative pronoun) has become stressless; in sense 5 it has disappeared, and ‘he once lived here’ appears as the direct object of ‘we know’.

After aware, certain, conscious, suspicious, assured, informed, persuaded, etc., of or some other preposition seems understood before that:

‘I am certain of that: he once lived here’. But ‘I am certain that’ may have arisen as another way of saying ‘I know that’; and so of the other expressions.]

1873 J. Morley Rousseau I. vii. 284 Rousseau was persuaded that Madame d'Epinay was his betrayer.

“While some might think that all professors do is lecture a few times a week and assign some homework, that is not all, Prof”.

As you see, this falls into (4) above, and can be omitted as per (5).

The second “that” is a pronoun with the referent of “what I have just said”, or the whole of the previous clause.

As with most things in modern spoken English, context is everything. I, for instance, could say "That is a very good question." and we would all know I am referring to the question asked in this post, but I just said, "That's the best answer." I may leave the readers wondering to which answer I was referring.

There are common phrases used in English such as: "That's that." which is an entirely complete thought and sentence meaning something has come to an end and now it is known in its entirety... there is no more to be said or learned about the matter.

Also such as: "That he walked back and forth on the same blocks for 8 hours daily remained a mystery for many years." where the action "that" refers to is actually described after the pronoun appears in the sentence.

English is weird, but it's all I've got.