Does "I like my new car" express a permanent or temporary state?

The difference between present continuous and simple present is a matter of degree rather than of absolute permanence: "I like my new car" is a more permanent condition than "I am liking my new car" which expresses the idea that it’s something happening around now, as you pointed out yourself, so the usage "I like" is appropriate in this case: I expect you will continue to like your new car for quite a while which is what is expressed by 'more permanent' here.

Whereas 'I am liking' gives a subtly different impression although it is idiomatically sound and even informally typical usage, especially for some native speakers. The famous slogan for McDonald's says, "I'm loving it!"

This is the subject of an earlier question here at English.SE: How normal-sounding is the slogan "I'm lovin' it" to native ears?

Grammar Girl discusses whether it is 'proper grammar' here

http://www.quickanddirtytips.com/education/grammar/is-im-loving-it-proper-grammar

and gives this notable ruling:

Idiomatic Uses of Stative Verbs

According to the rule, “I’m loving it” is not grammatically correct because it uses a stative verb—in this case, one that conveys emotion, love—in a progressive tense.

But, now we come to some idiomatic uses of stative verbs. You can conjugate certain stative verbs in a progressive tense in the right context [...]

The Verdict

“I’m loving it” does sound slightly off, and that draws attention. Perhaps that’s why McDonald’s chose it for their slogan [...] We all know that advertisements, song lyrics, and fashion headlines aren't the places to turn for examples of good grammar, but we also know that native speakers of English can get creative with traditional grammar, and that sometimes grammatically iffy phrases catch on. Language is constantly changing. Enough people seem to be using stative verbs in progressive tenses that we can probably say it’s becoming more accepted in popular culture to use them that way.

In your case 'I'm liking it (my new car)' would imply a more transient, living-in-the-moment feeling, with very much a McDonald's vibe, giving a sense of how you are really enjoying the newness and the whole feeling of having a 'new' car.

However, if by 'permanent' you are implying that you may possibly no longer like your new car once it becomes older, may I suggest the original sentence would have 'semantically expired' by then -- as in, your new car would have become your 'no longer new' car by that point and the statement about liking it would therefore no longer be applicable IMHO.

From the site https://www.ecenglish.com/learnenglish/lessons/non-continuous-verbs

It is important to understand that not all verbs can be used in the continuous form. We will refer to these verbs as ‘Non-continuous Verbs. Another thing which is particular about verbs and the continuous forms is that some verbs that can be used in both the simple and perfect forms and also in the continuous forms have two different meanings.

Verbs that are physical actions which you can see someone doing can be used in all forms and with all tenses; run, walk, eat, read, fly, say, touch etc. can be used in all tenses.

Non-continuous verbs; verbs that cannot be used in continuous forms are usually verbs that you cannot see somebody doing. These verbs are rarely used in continuous forms. They are: Abstract verbs Be, want, cost, need, care, contain, owe, exist etc. Possession verbs Own, belong, possess etc. Emotion Verbs Like, love, hate, dislike, fear, envy etc.; She needs help. Not She is needing help. He wants a break. Not He is wanting a break.

Thus, you don't normally use "I am liking," even when we're talking about a short period of time.

This site http://www.englishtenses.com/present_simple_table shows that most stative verbs, when used in the continuous tenses, have other, more active meanings (to see = to sense with the eyes; to be seeing to have an appointment or to regularly go out on dates with someone). But the distinction they give for to love/be loving seems a little arbitrary to me, just to make it fall into line with the others. This site http://www.perfect-english-grammar.com/stative-verbs.html mentions some stative verbs that can be used in the continuous tenses with a meaning shift, but doesn't include to love.

As regards the page you cited, it's true that sometimes people say things like "I'm liking/missing/loving (something/one)", but it's very informal and most people would not consider it "good usage," although McDonald's may beg to differ:

https://www.youtube.com/watch?v=LCcHrDMsfDg

But note the apostrophe in "lovin'", which is another marker of informality.

If you do use the continuous form for stative verbs, I don't think you're conveying any degree of temporariness, only informality. With other, non-stative verbs (she lives/is living with her parents), the temporary/permanent difference is there.

By the way, I might say "I'm missing my keys" because I can't find them. "I miss my keys" would be weird, because that would imply that they're gone, and I had an emotional attachment to them.

So, if I said it grammatically correctly "I like my new car", would people think that that sentence expressed a permanent or temporary state?

For stative verbs such as like, the simple present is a neutral form, not implying anything about permanence or temporariness.

If you want to emphasize that your feeling is likely to be temporary, you can preface your statement with "right now".

If you want to emphasize that your feeling is likely to be permanent, I think your best bet is to add something that ties your feeling to something about it: "I like my new car much better than my old one", or "I like my new car; its all-wheel drive is exactly what I need on snowy days", or . . . you get the idea.

The first thing to get clear is that the idea of 'permanent' and 'continuous' or 'temporary' are ways to express quite a tricky idea. Almost any such words give at best a rough idea what is going on.

Suppose we divide verbs into those that describe actions or events (I walk, I drive, I eat....), and those that refer to states of affairs (! know, I like, it hurts...

The distinction between so called 'more permanent' and so-called 'more temporary' seems at first sight obvious, especially because of the cautious qualification in the use of the comparative 'more'. But the easiest way of looking at it is to say that the the continuous present is referring to an event or action that is taking place at the time at which the sentence is uttered.

1.1 She is driving to Newcastle 1.2. My company is conducting a review into its internal efficiency 1.3 The level of the Thames is rising at such an alarming rate that within house it may break its banks and flood the streets.

We are talking about events that are physically taking place now. But they are not 'temporary' or 'more temporary', so much as references to things actually taking place at the time of writing or speaking. That is the function of the 'present continuous' in relation to verbs of this sort. So from your question it is clear that you know the difference between 'I play the flute' and 'I a playing the flute. In French, of course, 'Je joue de la flute' can only be made fully univocal by reliance either on context or on phrases like 'Je suis en train de jouer de la flute.' ('I am in the process of playing the flute').

So the so-called 'simple' present refers to what is the case in another sense: stable states of affairs over some sort of long term. These are works that refer to states of mind or body or about the way things are: words such as ''! work in the factory down the road', or 'I agree with you', or 'the tide rises and falls at regular intervals under the influence of the Moon.'

The problem is that states of mind notoriously blur the boundary between the two kinds of present.

2.1 I know the square root of sixteen. I suppose I might forget this piece of information at some time. But to all intents and purposes, this kind of state is perpetual. It is not something I am doing now. Moreover, we recognise 'I am knowing the square root of sixteen' as a sign that the person saying that is not a native speaker of English. Moreover, this kind of knowledge is not something I am aware of all the time that I know it. It it not a conscious presence to me. I do not walk around thinking about it.

However, there are some such situations, where speakers seem to feel as if they are in the middle of the grey area.

2.2 Do we say "I suffer from a chronic illness.", or "I AM SUFFERING from a chronic illness."? Well, we seem to feel pulled in both directions. The rule doesn't help us. The chronic illness is an established and continuing state, so perhaps the simple present should be used, but a chronic illness is surely a something that is happening to me and so perhaps the continuous present should be used. There is no answer.

And then there are your example of apparent exceptions.

2.3 She lives or is living with her parents. You could draw some distinctions here. With the simple present, it does sound (to me, subjectively, as if this means that this circumstance is certainly long term. With the continuous present it looks more like a temporary arrangement, while she completes a PhD, or is getting her life back together after splitting with a partner. But we are in the grey area.

2.4 I like or am liking my new car. Here I should say that the use of the continuous present is a fairly recent and special usage. It is more like saying "I am enjoying or relishing my new car". It is, in fact, something I am experiencing it at least until I am fully ised to this magnificent automobile. But this will fade with custom, even though I shall not like it any the less - till it gets old and starts needing costly repairs or is displaced in my mind by a newer and more desirable model.

2.5 I'm missing (or I miss) you. This is, I think, bang in the middle of the dilemma. I can miss someone in either of these two ways. Because Missing someone, though not, perhaps, felt literally constantly, is felt often enough in a day to count as something that is happening to me. And yet it can equally be seen as a steady state, part, if you will, of my settled feelings about the person. So continuous and simple presents are equally possible.