The reason why temps is invariable and always has an s is because of its etymology. It comes from the Latin tempus¹.

Temps follows the general rule for liaisons, I do not think there is a case where it can be mandatory, you can say : /detɑ̃imemɔʁjo/ or /detzɑ̃imemɔʁjo/ (des temps immémoriaux).

Temps is invariable but it can be either be singular or plural:

When it expresses duration it is singular.

• Le temps passe vite. (Time flies)
• Je n’ai pas le temps.
• Une montre sert à mesurer le temps.

Obviously if you talk about mutiple durations, then temps is countable and can be used in the plural as
Gregorycrene mentions in their comment.

• On note les temps mesurés pour chaque expérience.

When it means the right period of time to do something, it is singular.

• Aout est le temps des moissons.
• Le temps des cerises

When it means an era it can be plural.

• Les temps sont durs.

But it can also be singular :

• Par les temps qui courent or par le temps qui court (In this day and age)
• C’est fini le temps où je pouvais courir 10 km sans m’essouffler.

_{This answer most probably does not encapsulate the whole issue about temps.}

_{¹Note that in Old an Middle French the word was spelt tens or tems, the p was reintroduced in classical French.}

I

Your question requires really a vast answer of which you are not going to be given but a few little parts.

The term for the quantity is “le temps” ou “la durée” in many cases; so, mainly you do translate “time” by “temps” but there is a great number of exceptions in the way of idiomatic expressions, and, as usual, you’ll have to learn those case by case. In certain context, of which the time as you read it off a clock is but one (“heure”), another word is used.

• (time on/off the clock, point time) What time is it ? : Quelle heure est-il ?
• (music) in time: en mesure
• (points in time, in general) This time it happened. : cette fois c’est arrivé.
                                             It wasn’t the right time. : Ce n’était pas le bon moment.
  …

“Temps” ou “durée” is the term you use in physics, in sports.

In sports you use regularly “temps”, not “durée, when talking about a performance from the point of view of classifying it, that is from the official point of view or the record keeping matters.

• Le marathon a été couru en un temps de moins de deux heures.
• Oui, cela veut dire que pendant une durée/un temps d’environ deux heures le coureur n’a aucun répit. (“durée” rather than “temps”, ngram)

In physics you use only “temps” in certain contexts, for instance

• in dimensional analysis (ML/T²),
• in talking about the physical unit (Le temps est une des grandeurs de la physique.)

However, even while talking professionally about physical phenomenas you can use “durée”.

• Sur toute la durée de la trajectoire il existe une force F sur l’objet.
  (Over the whole time of the trajectory, a force F is being applied on the object.)

II

• Time flies ! => Le temps passe vite ! (Apparently there is no idiom and you can take it up upon yourself to render the English idiomatic expression by something of your invention.)
• Le temps passe à une allure folle. • Le temps passe à toute allure. • On ne sent pas le temps passer. • Le temps passe vite.

III

• Time is an illusion. => Le temps n’est qu’une illusion.
• Time is what our clocks measure. => Le temps c’est ce que nos (montres et) horloges mesurent.

The letter “s” at the end is just part of the spelling both in the singular and the plural and is never pronounced. The reason for it is not given in current sources but let’s say that it’s been found as part of the word as far back as the middle of the 10th century (950).

(TLFi) a) 2e moit. Xe s. avec dém., désigne une période du passé (St Léger, éd. J. Linskill, 13: a ciels temps);

When you use the plural word and that you are not talking abouts several points in time or quantities of time, that is measured time, you are talking about another concept than physical time, and that is the concept of historical period at large (une époque vague, imprécise), which, by the way, can be situated in the future. (By plural word is meant the word for which a plural verb is necessary, as the s at the end of “temps” is not an indication of number.)

• Les temps étaient durs, mais tout passe, la guerre aussi.
• Quand ces temps viendront il n’y aura plus lieu de se demander s’il y a quelque chose à faire.

There are numerous idiomatic expressions making use of this plural form.

• (TLFi) Les temps actuels, bibliques, fabuleux, historiques, modernes, mythiques, préhistoriques, autres temps, autres mœurs (à chaque époque, un mode de vie différent)
  les premiers temps, les derniers temps, ces temps derniers, dans la nuit des temps.
  dans les premiers temps. (au commencement)

There are exceptions; for instance, “être dans les temps” in various sports means that the time that has been measured at a certain point of a competitor’s performance is about what should be expected. You always use the plural form then, and that is the word “temps” as the physical quantity.

IV

On ne fait pas cette liaison d’habitude ; je ne l’entends pas dans le parler des gens et personnellement je ne la fais pas.

dans les temps à venir /dɑ̃ le tɑ̃ a vəniʁ/