So, Can linking verbs be used with "continuous tenses"?

The [finite form of to be] + present participle is used not only to denote continuous states or progressive actions but also informally, I submit, the ingressive or inceptive aspect, i.e., that an action or state has begun. Neither statal nor copulative/pseudo-copulative verbs are always exempted, especially in informal speech. There are other uses which have nothing to do with aspect but are used as intensifiers and polite hedges.

Some forty minutes into an hour baking time for fresh bread, the Malliard reaction is doing its magic:

That bread is smelling good.

I am not suggesting that the bread is in a continuous state of smelling good, but that the bread has begun to smell good (ingressive aspect) and it's time to get the butter out of the fridge, because I'm fixing to have a big slice of homemade bread (prospective aspect, but only for some speakers of Southern American English).

In a more formal register, I would say:

The bread is beginning to smell good and I'm about to have a slice of homemade bread.

Think of any food that looks unappetizing before cooking — steaks on the grill, a stew simmering for hours, dried beans — and wait for The Moment.

By like token, I could be mixing the hot and cold water to get just the right temperature for a bath or shower:

The water is feeling nice and warm now. The water has begun to feel nice and warm now.

This apparently meme-worthy citation describes an emergent feeling of loneliness or contingency upon contemplating the vastness of the universe (ingressive aspect) using the meaning of seem as "seem to feel" and in the continuous form, accentuating a moment of intense emotion right now. It invites the reader to identify with the author in a way which to her apparently wouldn't be as rhetorically powerful as simply saying the universe seems huge. The universe, of course, has been huge since the Big Bang and it always is, but in this moment, she has begun to feel it.

At work he's a total jerk. He is being a total jerk [today at work].

This usage is best solved by acknowledging a lexical difference: in the present tense one is making a judgment about his personality; in the continuous, instances of behavior, in this case, today.

Permanet states or non-animates without agency can never be used with the continuous:

*She is being Australian. (nationality, not in some odd scenario behavior)

*He is being short.

*My shirt is being green.

You are looking pale today. You are looking fabulous today.

While the first sentence could be parsed as ingressive, this usage seems more like a polite hedge, while the second is best viewed as an intensifier which, like the meme, suggests a heightened level of personal involvement.

Can linking verbs be used with “continuous tenses”?

I think the answer is: yes, at least in some cases.

I'm really not sure what to tell you; some of the examples online of supposedly incorrect formations sound fine to me (a native speaker).

• I’m feeling a little better today.

This is perfectly idiomatic (the definition you found does not say otherwise, either). You can find plenty of examples in this Google Books search.

• The water is feeling warm.
• Its skin is feeling really smooth.

These are modified versions of the examples marked as "not used in the progressive tenses", which still seem idiomatic to me. See here and here for some examples (not as many as the first search but still plenty).

• You are looking good.

Another perfectly idiomatic utterance. See this ELL question for details. "It's looking interesting" is also idiomatic, as can be seen in the number of results here.

• *He is seeming happy.
• *He is appearing happy.

Neither sounds right to me. For the first there are essentially no hits on Google Books except for false positives and grammar books telling you what not to say. For the second there are just too few hits on Google Books.

• I am being silly.
• *I am being tall.

The first one is idiomatic, the second one is not. These examples are from this site, which explains:

Kolln suggests that we think of the difference between stative and dynamic in terms of "willed" and "nonwilled" qualities. Consider the difference between a so-called dynamic adjective (or subject complement) and a stative adjective (or subject complement): "I am silly" OR "I am being silly" versus "I am tall." I have chosen to be silly; I have no choice about being tall. Thus "tall" is said to be a stative (or an "inert") quality, and we cannot say "I am being tall"; "silly," on the other hand, is dynamic so we can use progressive verb forms in conjunction with that quality.

The same applies to verbs. Two plus two equals four. Equals is inert, stative, and cannot take the progressive; there is no choice, no volition in the matter. (We would not say, "Two plus two is equalling four.") In the same way, nouns and pronouns can be said to exhibit willed and unwilled characteristics. Thus, "She is being a good worker" (because she chooses to be so), but we would say "She is (not is being) an Olympic athlete" (because once she becomes an athlete she no longer "wills it").

A link verb does not indicate an action or a state - it simply links or attributes information in the complement phrase (a noun or an adjective) to the subject. The true link verbs are be, become and seem. Then there are the quasi link verbs like feel, appear, look, taste, etc.

Be, when it cannot be substituted with exist and is followed by a noun or adjective complement, is a link verb. Become and seem are practically always link verbs. None of these uses indicate an action or state. One test for link verb is if you can substitute the quasi link verb with a true link verb, then you've got a link verb (but always followed by a noun or adjective complement).

Feel is not a true link verb because it so often indicates the action of feeling or experiencing something. How a construction that uses these kinds of verbs actually functions is often a matter of the complement type and interpretation.

I feel hunger. S+V+DO - feel is an action verb; present simple; statement of fact (not routine). (People don't say this when they are hungry, btw). Try the substitution - you can't do it, it's not a link verb.

I feel hungry. S+LV+Adj - present simple - feel is a link verb (adjective complement), same as I am hungry.

As a link verb feel doesn't take a noun as a subject complement. Feel, even as a link verb retains some of its active aspects. To feel hungry, means to feel your hunger. In I am hungry the verb is not at all active. I think most of the quasi link verbs behave in this lopsided way.

In the case of using continuous tenses with link verbs, I'm not entirely sure but I think a sentence like:

I am feeling hunger. S+V+DO is grammatical (but people don't say this when they are hungry) but I am feeling hungry is not technically grammatical because 'am feeling' as a continuous action verb needs an object complement.

Nevertheless, I am feeling hungry or The cake is looking delicious are acceptable idiomatic constructions, but are harder to pin down grammatically. I think I am hungry and The cake looks delicious are better choices with the exact same meaning because they are clearly grammatical.

So, if you want to see a link verb in such a construction, you might have to look at it this way (I'm not sure about this):

I am feeling hungry. S+LV+Noun group (gerund + post position adj)

Where 'feeling hungry' functions as a noun complement of the LV 'be'.

Or perhaps one could see feeling hungry as an untensed verb phrase/ participle functioning as the subject complement.

S+LV+N

A similar explanation might suffice for, 'You are looking good.'