What is the opinion of Buddhism on the caste system and untouchability of Hinduism?

In your question, you have performed what is called the logical fallacy of 'False Equivalency'.

The caste system in Hinduism as practised in India and Nepal is a decree of the Hindu scriptures. It is mentioned in Hindoo scriptures like Gita and Manusmiriti. So the professions in India have originated from the Caste of a person. Moreover, the Hindoo caste system tells you that if you are a Dalit then you have to do good karma to first become a Brahmin and then you can attain the moksha (as per Manu Smriti).

In the examples you have given, it is more like a class system taken to an extream. That was originated as a discrimination in the society and not handed down by Buddhism. Buddhism as such has nothing to do with that inequality. All the people if Buddhist in these countries are on an equal footing of Dhamma and not discriminated by the Dhamma.

As we know as Buddhists that the Buddha was no hindoo, then why this system is also in the societies of Buddhist countries?

Again, if a hindoo follows their Dharma strictly then they have to adhere to the caste system.

And if a Buddhist will adhere to the Dhamma strictly then He/She will NOT adhere to any discrimination.

So, why is this still practised in some Buddhist countries?

Its because people are not well instructed in Dhamma. People there are Buddhist just for namesake because their parents are Buddhist or the society is Buddhist. The Dhamma is not practised by them, that way.

I cannot agree with the following i think everyone followed caste system it is because of westernization and modernization that they are disappeared and by the way division of society shall remain dalits also discriminate against mahadalit the only way to fight such inequalities is to follow sardar vallabhbhai patel

The most important idea in Buddhism with regards to caste is that regardless of which background one is from, if one is righteous, consummate in virtue, and earnestly seeks the path to the end of suffering, he would definitely find it.

As to why certain Buddhist majority countries have some kind of caste system - well, that does not come from Buddhism. It comes from the natural tendency of human societies to form socioeconomic clusters, which applies to all countries in the world, including the fully developed countries. Please see this article on inequality in the USA.

However, what is special in Hinduism is that the caste system is found in the religious scriptures and is part of the religion. A religion-enforced caste system is not found in Buddhism. Buddhism also does not comment on the political ideology of lay societies e.g. democracy, capitalism, monarchy, socialism, communism etc.

From the Gihi Sutta (AN 5.179):

In the same way,
wherever one is born
among human beings —
noble warriors, brahmans,
merchants, workers,
outcastes, or scavengers —
if one is tame,
with good practices,
righteous,
consummate in virtue,
a speaker of truth,
with conscience at heart,
one who's abandoned birth & death,
completed the holy life
put down the burden,
done the task
fermentation-free,
gone beyond all dhammas,
through lack of clinging
unbound:
offerings to this spotless field
bear an abundance of fruit.

From the Sundarika Sutta (SN 7.9):

Then Sundarika the brahmin went up to the Buddha, and said to him: “Sir, in what caste were you born?”

“Don’t ask about birth, ask about conduct.
For any wood can surely generate fire.
A steadfast sage, even though from a low class family,
is a thoroughbred checked by conscience.

Thought, and food for such:

There is not really any evidence that the Buddha ever thought on such as "social reforms" and no where he actually denied the difference of peoples social status. He "only" explained it as a matter of kamma, deeds, whether one might be born in a good family or not, and that one to be called, become this or that, being a matter of ones deeds and not something inherent, gained by birth, or being given by someone.

People who believe in Kamma (even if not total understanding, not seeing escape), whould be the first to be accepted in his community. So it's merely unlikely that the good Dhamma, not corrupt, could ever appear in regions where people believe that all are equal by birth, have inherent equal capacities...

What ever region, society, has not fallen into the ideas that beings are equal, going after cutting all down to the lowest equal level (marxistic, pseudo-liberal), still has certain right view and respecting differences, will care certain casts on, an may it be between self-employers, professionals, workers... or what ever.

People of different social background, virtue... will always gather together according to their tendencies. Even there where it's different declared as "should be such and such". And it's naturally that the most wise are the first to disappear, or will be ever very small groups.

Or has one traced any society which has not it's "casts"? Even thought the Buddha told that the best dwelling would be accordingly to ones attainments, a sotapanna equal a sotapanna... Arahat equal an Arahat. Even his hierachy points out classes very clear, yet not by birth or as something given, of course.

What ever the Buddha had taught was never thought to use it as politics or to design worldily societies and at least it's a matter of Upanissaya, strong condition causes (vy kamma) where one appears, takes birth, tended to. If one seeks for change, then only ones own work, own sacrifices, will have long term effects. Nothing really gained when wishing to pull those in current better situation down to ones level.

So this path of liberation is open for everyone, everyone willing to leave house, his stand, which goes along with the lose of the low fetter of group(identification)view.

Becoming really Buddhist and holding on house, group ones society wouldn't really work. It's simply hypocritical if thinking to holding on both.

(Not given for trade, exchange, stacks, entertainment, what ever binds to this going on, but for liberation, encouragement for own skilfull work and letting go)

Actually, Gautama Buddha was not against the caste system in mainstream non-Buddhist Indian society however he did emphasize social status ('caste'/'jati') was dependent upon here-&-now kamma rather than on physical & social birth (refer to MN 93, MN 98 and other suttas in the Brāhmaṇa Vagga). For example, he often emphasized a Brahman priest or caste member that does not adhere to the five moral precepts is not a Brahmana or Holy Man.

The Pali suttas have the following examples:

1. It was declared the Buddha "does not seek any harm for the line of brahmins" (MN 95).

2. The Buddha criticised Brahmans who marry & have sex with non-Brahmans. (AN 5.191)

3. The Buddha said wives who are immoral are expelled or caste out from families (MN 37.30).

The Pali suttas show no opposition to the non-Buddhist Indian caste distinctions however the Pali suttas do explicitly say when a person joined the Buddhist Community (Sangha) they automatically lost their former caste status or name (AN 8.19).

(4) “Just as, when the great rivers … reach the great ocean, they give up their former names and designations and are simply called the great ocean, so too, when members of the four social classes—khattiyas, brahmins, vessas and suddas—go forth from the household life into homelessness in the Dhamma and discipline proclaimed by the Tathāgata, they give up their former names and clans and are simply called ascetics following the Sakyan son. This is the fourth astounding and amazing quality that the bhikkhus see in this Dhamma and discipline….

As for contemporary Asian societies, many of these societies were previously or concurrently Hindu in relation to the arrival of Buddhism. Similar to when the Buddha was alive, Buddhist monks generally do not act to oppose & destroy other religions. Buddhism is not like Judaism or Christianity. Thailand is an excellent example, where most religious people still follow other religions, such as spirit, ancestor or Brahma worship (even though they consider themselves "Buddhist").