I know that expect is used this way:
I expect you to do that.
But I have also seen examples like with verb in its "ing" form:
> What to expect working at... > I will expect you doing //sounds not right to me
I would be grateful for explanation.
The verb to expect always takes an infinitive complement:
This is the normal way of forming sentences with expect. Your last example above is, as you suspected, grammatically incorrect:
The other example that you gave with expect + -ing is something else. In this case., the verb working is the beginning of an adverbial phrase, and isn't a complement to expect at all.
In this sentence what to expect is a nominal relative clause indicating what you should expect, and working at BigCorp is an adverbial gerund indicating where you will be working.
Other examples can be constructed that put expect and a gerund together, but in all of those examples the gerund is not a complement to expect, but an independent grammatical phrase which just happens to come close to expect. The actual complement of expect is always an infinitive.
or with a different NP appearing as subject in the complement
But it can't occur with a Gerund complement:
What's going on with
is that the gerund clause is not a complement (Noun) clause, but a reduced Adverb clause, which can appear in many places in a sentence, but which happens to appear here right after expect, exactly where a complement would appear. Score one for English syntax.