Confusing structures with modal verbs

Solution 1:

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.

Solution 2:

1) I should / I shouldn't to give somebody advice:

• “Is it cold?”
  
  “Yes, I should wear a coat.”
  
  (It is not a misprint! “I” refers here to another person.)
  
  
• “I shouldn't stay up too late: you'll be tired tomorrow.”

Using “I should” in the first person, corresponding to the present-tense form “I shall”, is of a more formal register than occurs in most pedestrian conversations. The normal exchange is just:

“Is it cold out?”

“Yeah, I’d wear a coat.”

Which neatly avoids the entire question. If you are forced to expand the otherwise-ubiquitous contraction, would becomes the normal form in regular conversion:

“If I were you, I would wear a coat.”

See this wikipedia article for more.

Note however that this form does nonetheless still occur in certain constructions. For example:

“It’s going to be 20 below tonight. Do you think I should wear a coat?”

“I should say so!”

Solution 3:

6) Begin the sentences from 5) with should:

• “Should Tom phone, tell him I’ll call him back later.”

English hypotheticals do not require an if or an unless. Were it not so, you would have been informed of this fact. Should it ever change, we shall send you a cable informing you of the event.

In this instance, you have stumbled upon OED sense 21a for shall.

21 a. In a hypothetical clause relating to the future, should takes the place of shall (indicative or subjunctive), or of the equivalent use of the present tense, when the supposition, though entertained as possible, is viewed as less likely or less welcome than some alternative. (With future, future perf., or imperative in the apodosis.)

Two of the OED’s citations for that sense make clear that this is what they are talking about:

• 1846 J. Baxter Libr. Pract. Agric. (ed. 4) I. 50
  Should any soluble salt remain it will be soda.
• 1896 A. Austin England’s Darling i. iii,
  And, should the looked for shock be on us soon, I must be there!

It’s a simple conditional. If you are asking about register, it is a bit formal, but hardly uncommon.