Новая грамматическая тема: конструкция «лучше + инфинитив» помогает давать совет мягко и спокойно.

Because they both describe тема, and тема is a feminine singular noun in the nominative case.

Russian adjectives must agree with the noun they describe in:

• gender
• number
• case

So:

• новый → masculine
• новая → feminine
• грамматический → masculine
• грамматическая → feminine

Since тема is feminine, we get:

• новая тема
• грамматическая тема

The colon introduces or explains what the new grammar topic is.

So the structure is basically:

• New grammar topic: ...
• then the sentence tells you what that topic is.

This works very similarly to English punctuation in a heading-like sentence.

This construction means something like:

• it’s better to...
• you’d better...
• it would be better to...

In Russian, it is often used to give advice in a soft, calm, non-pushy way.

Examples:

• Лучше подождать. = It’s better to wait.
• Лучше не спорить. = It’s better not to argue.
• Тебе лучше отдохнуть. = You’d better get some rest.

So in the sentence, the point is that this construction helps you give advice gently.

Because this pattern is built that way: лучше + infinitive.

The infinitive expresses the action being recommended.

Examples:

• Лучше уйти сейчас.
• Лучше спросить учителя.
• Лучше не торопиться.

You are not saying who is doing the action through the verb ending. The construction stays more general and softer that way. If needed, you can add the person separately:

• Мне лучше пойти домой.
• Тебе лучше подождать.
• Вам лучше позвонить заранее.

No. You can use лучше + infinitive in two common ways:

1. General advice

   • Лучше не спешить.
   • Лучше сначала проверить.

2. Advice for a specific person

   • Тебе лучше отдохнуть.
   • Вам лучше обратиться к врачу.
   • Мне лучше промолчать.

When you name the person, Russian usually uses the dative case:

• мне
• тебе
• ему / ей
• нам
• вам
• им

Not in this construction.

In лучше + infinitive, лучше means it would be better to..., and you do not need чем.

Example:

• Лучше подождать. = It would be better to wait.

But in a real comparison, лучше can mean better than, and then you may use чем:

• Этот вариант лучше, чем тот. = This option is better than that one.

So there are two different uses:

1. Advice

   • Лучше уйти сейчас.

2. Comparison

   • Этот план лучше, чем предыдущий.

After помогать (to help), Russian often uses an infinitive to show what action is being helped.

So:

• помогает давать совет means
• helps [someone] give advice

This is a very normal structure in Russian:

• Музыка помогает расслабиться. = Music helps [you] relax.
• Эта схема помогает понять правило. = This diagram helps [you] understand the rule.
• Конструкция помогает говорить вежливо. = The construction helps [you] speak politely.

Because давать is the imperfective verb, and here the sentence is talking about a general ability or usual function, not one completed act.

Compare:

• давать совет = to give advice / to be giving advice / to give advice in general
• дать совет = to give one piece of advice as a completed action

In this sentence, the meaning is general:

• the construction helps you give advice as a type of action

That is why давать sounds natural here.

Russian often uses the singular to talk about something in a general sense.

So давать совет here means:

• to give advice
• to give a piece of advice
• to offer advice

This is similar to how English can say give advice in a general sense, even without focusing on one specific piece.

Also, давать совет is a very common collocation in Russian.

Лучше + infinitive is usually softer and sounds more like a suggestion or calm advice.

Compare:

• Тебе лучше отдохнуть.
  = You’d better get some rest.
  Soft advice.

• Тебе нужно отдохнуть.
  = You need to get some rest.
  Stronger, more direct.

• Тебе надо отдохнуть.
  = You need to rest.
  Also fairly strong and practical.

So if you want to sound less forceful, лучше + infinitive is often a good choice.

Because they describe how the advice is given.

The sentence says the construction helps give advice:

• мягко = softly / gently
• спокойно = calmly

These are adverbs, not adjectives, because they modify the verb phrase давать совет.

Compare:

• мягкий голос = a soft voice
  adjective describing a noun

• говорить мягко = to speak softly
  adverb describing an action

And:

• спокойный человек = a calm person
• говорить спокойно = to speak calmly

Yes, very often. This is one of the most useful things about the construction.

Examples:

• Лучше не спешить. = It’s better not to rush.
• Тебе лучше не говорить об этом. = You’d better not talk about that.
• Вам лучше не опаздывать. = You’d better not be late.

The usual pattern is:

• лучше не + infinitive

This is a very natural way to give polite negative advice in Russian.