Learn English – “How do we call (something) in English?”

Purely in terms of the terminology, then a distinction sometimes made is that:

• inflectional subjunctive is the type found in Old English, German, modern Romance languages etc, in which verbal inflection distinguishes subjunctive from indicative;
• periphrastic subjunctive is the type found (if you adopt this analysis) in modern English, in which subjunctive is distinguished from indicative by way of modal auxiliaries/other verbal constructions.

There's really no consensually agreed upon "wrong" or "right" answer to the question of whether English actually "has a subjunctive". If you adopt the analysis that "subjunctive" is the grammaticalisation of non-assertive force with a verbal paradigm-- which seems to be a close approximation to what the phenomenon is in Romance languages-- then it's fairly clear that English doesn't have such a phenomenon. (Saying that English has a past subjunctive on the basis of the form "if it were" is a bit like saying that English is a verb-final language on the basis of a phrase such as "Language does not a society make": it's proposing a paradigm on the basis of a rare exception.)

If you extend the definition to cover cases such as English "It is sad that he should leave", "David commanded that she leave" etc, then there are various issues to be considered which are typically glossed over in language learning textbooks:

• care must be taken to recognise where the similarities and differences actually lie between these phenomena and the inflectional subjunctives of French etc;
• it is worth thinking about what the motivation is for proposing "subjunctive" as a 'special case' of modal verb usage.

3) will for present habits:

• “Every morning I will get up early.”

• “I’ve tried everything — the car just won’t start.”

There is no question that the verb (or verbs; there may be several) will is one of the very trickiest ones in the English language for foreigners ever to master. The deontic senses are seldom intuitive to a non-native speaker. I strongly advise you to carefully study the OED’s entry for this word’s incredibly many subtle uses.

In this case, your two examples are not of the same thing at all, and you have mischaracterized them. The first uses will to express habitual action; it does not indicate a simple future situation. This is the OED’s sense 8 for this verb:

8. Expressing natural disposition to do something, and hence habitual action: Has the habit, or ‘a way’, of ––ing; is addicted or accustomed to ––ing; habitually does; sometimes connoting ‘may be expected to’

This is related to sense 15, which is still not a simple future, albeit perhaps closer to that:

15. As auxiliary of future expressing a contingent event, or a result to be expected, in a supposed case or under particular conditions (with the condition expressed by a conditional, temporal, or imper. clause, or otherwise implied).

Your second example, the one about the car, is completely different. This corresponds to OED sense 12:

12. With negative, expressing the contrary of senses (def#6), (def#7), (def#10), (def#11): thus commonly = refuse or decline to; emph. insist on or persist in not --ing. Also fig. of a thing. (See also (def#9), (def#13).)

Here, your car is persisting in not starting. It is the figurative sense at the end extending to things, as though they had the will to refuse. The referenced senses 9 and 13 are respectively:

9. Expressing potentiality, capacity, or sufficiency: Can, may, is able to, is capable of --ing; is (large) enough or sufficient to.

15. As auxiliary of future expressing a contingent event, or a result to be expected, in a supposed case or under particular conditions (with the condition expressed by a conditional, temporal, or imper. clause, or otherwise implied).

As I said, will is quite complex. Please study standard reference works regarding its use.