Expect +to VS expect + ing

The verb to expect always takes an infinitive complement:

I expect it to rain.

They expect us to present tomorrow.

This is the normal way of forming sentences with expect. Your last example above is, as you suspected, grammatically incorrect:

*I will expect you doing well.

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.

Here's what to expect working at BigCorp.

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.

Expect can occur with an Infinitive Object Complement, either with an Equi subject:

• A Configuration: Bill expects [for Bill] to do that. ==> Bill expects to do that.

or with a different NP appearing as subject in the complement

• B Configuration: Bill expects [for] Mary to do that. ==> Bill expects Mary to do that.

But it can't occur with a Gerund complement:

• *Bill expects [Bill]'s doing that. ==> *Bill expects doing that.
• *Bill expects Mary('s) doing that. ==> *Bill expects Mary('s) doing that.

That's why *I will expect you doing ... sounds bad to you. It is bad.

What's going on with

This is what you should expect working at SlimeCo.

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.

• This is what you should expect, [while you are] working at SlimeCo.
• [While you are] working at SlimeCo, this is what you should expect.