Learn English – Modal verbs in reporting

I think you are misunderstanding the meaning of finite. Finite is not a category of verb but a category of verb forms and uses.

Finite forms are those which must take either past or non-past tense (must is anomalous in having the same form for both tenses) and may change to agree with the person and number of their subjects. Non-finite forms, infinitives and participles, do not change with tense of the utterance or person or number of the subject.

In your example, the modal verb could is finite: it is the "past" form of the verb can. (It may not express past tense, but that is another matter.) As you say, it doesn't take a particular inflection which expresses person and number; but no English verb has a complete repertory of such inflections. You will find a little more information at this question.

In fact, the full modals can, may, must, shall, will differ from other verbs in being defective: they have only finite forms, no infinitive or participles.

Every complete clause has exactly one finite verb^✲; if there are more verbs strung together, then the first is finite and the rest are non-finite. Consequently, if there is a full modal verb it must be the first in the string.

^✲ Except in cases where two or more finite verbs are conjoined: I can and will do it. But these cases really express two or more clauses.

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.