Does any employee of entity1 that use the software is allowed to request the full source ?
No. As the link that Brandin provided above makes clear, moving copies around inside a single organisation doesn't constitute conveyance, and doesn't trigger those obligations that are contingent on conveyance, source provision being one such.
If entity1 is to distribute binary to entity2, are we fine as long as we provide source if entity2 ask for it ?
It's possible. GPLv3 s6 governs source provision when object code is conveyed, and the only sections that any longer allow the "written offer" are 6b, which requires the object code to be embedded in a physical product, and s6c,which doesn't apply here. I'd always parsed s6's physical product requirement as "firmware in a functioning device", but the text of GPLv3 says this includes "a physical distribution medium". So if you ship the binary to entity two on (eg) DVD, then yes, you can provide a written offer to provide source on request. This offer must be valid for three years, or longer in certain circumstances.
Who at entity2 can make such a request for source ? Any employee of entity2 using the software ? Or can it be restricted to "someone in charge" to avoid ?
According to s6b, anyone who possesses the object code is entitled to request the source. This is not affected by whether they got that binary via regular conveyance or an internal transfer; if they have the binary, they are entitled to request the source. Furthermore, you cannot control what they then do with that binary and/or source, so long as they honour their GPLv3 obligations.
If software is installed on dedicated VM at entity2 that people use with remote desktop, does that change anything regarding "distribution" to employees of entity2 ?
Not really. That original provision of the VM image was a conveyance. It is true that further copies made and distributed inside entity two aren't conveyances, but that doesn't affect any recipient's eligibility to make a s6b source request.
I don't really understand how the licence work in corporate environment
Essentially, don't use copyleft software if you don't want copyleft obligations. They won't always apply, but it's much simpler to assume they will.
Edit: you write that you will likely remove the relevant libraries in an attempt to make your binary not subject to GPLv3. This may work, but it may not; it's not as easy as it sounds. I recommend Do you retain copyright in a project even if all your code has been rewritten? for further reading.