Новая грамматическая тема: предлоги «согласно» и «вопреки» с дательным падежом.

Both согласно and вопреки require the dative case.

• согласно плану = according to the plan
• вопреки совету = contrary to the advice

This is the main grammar point in the heading: after these prepositions, the following noun or pronoun must be in the dative.

Because с means with, and after с in this meaning, Russian uses the instrumental case.

So:

• дательный падеж = the dative case
• с дательным падежом = with the dative case

Here, дательным падежом is instrumental because it depends on с, not because of согласно or вопреки.

Yes, in this use they are prepositions.

Russian prepositions are not always tiny words. Some are longer and come from other parts of speech historically. What matters is how they function now.

In sentences like:

• согласно закону
• вопреки правилам

they function as prepositions because they connect to a noun phrase and require a specific case, namely the dative.

You use normal dative endings. Some very common patterns are:

• masculine/neuter singular: often -у / -ю
  
  • план → плану
  • закон → закону
• feminine singular: often -е / -и
  
  • система → системе
  • идея → идее
• plural: often -ам / -ям
  
  • правила → правилам
  • ожидания → ожиданиям

So:

• согласно закону
• вопреки системе
• согласно правилам
• вопреки ожиданиям

Yes. Pronouns also go into the dative.

Examples:

• согласно ему = according to him / it
• вопреки ей = contrary to her / it
• согласно им = according to them
• вопреки нам = contrary to us

Very common dative pronouns include:

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

Yes. The biggest mistake is using the wrong case after them.

For example, standard Russian says:

• согласно приказу
• вопреки приказу

A form like согласно приказа is considered nonstandard.

So the safest rule is simple:
after согласно and вопреки, always expect the dative.

Not exactly.

согласно often sounds a bit formal or official, especially in writing:

• согласно договору
• согласно инструкции

You will often see it in documents, news, academic writing, and formal speech.

вопреки is also somewhat bookish, but it is quite common in normal written and spoken Russian too:

• вопреки ожиданиям
• вопреки советам родителей

So both are common, but согласно is especially associated with formal style.

Normally they go before the noun phrase they govern:

• Согласно расписанию, поезд отправляется в шесть.
• Вопреки прогнозу, погода улучшилась.

This is the standard pattern learners should use: preposition + dative noun/pronoun.

Yes:

With согласно:

• Согласно плану, мы начинаем завтра.
• Согласно закону, это запрещено.
• Согласно расписанию, урок длится час.

With вопреки:

• Вопреки совету, он поехал один.
• Вопреки ожиданиям, экзамен был лёгким.
• Вопреки правилам, они вошли без очереди.

These are good models to memorize because they show the basic pattern clearly.

They are similar in meaning, but the grammar is different.

• вопреки
  • dative
• несмотря на
  • accusative

Compare:

• вопреки совету
• несмотря на совет

Both can often be translated as despite or contrary to, but they are not grammatically interchangeable in form. If you use вопреки, remember the dative.

Memorize the two prepositions as fixed patterns:

• согласно + dative
• вопреки + dative

Then learn a few chunks by heart:

• согласно плану
• согласно закону
• вопреки правилам
• вопреки ожиданиям

If you remember them as ready-made combinations, the case choice becomes much easier.