You can/should only use the HP code if entitled and yes the hotel is allowed to ask you to show you are entitled to rate. If at checkin time hotel carries out said check and you are not entitled then
A)you are past room cancel time (day before)
B)hotel has your CC attached to your booking
C)hotel can now charge you a higher rate to checkin, repricing as high as on the day walkin rate . You would have no choice in the matter, upto you if you want the risk

Yes most company rates , including HP (which is a non commissionable rate to agents/OTA's) are qualifying rates earning both stay and points credit
(A very small minority of companies negotiate slightly more discounted rates in return for making rate not point earning)

Company rates in and of themselves do nor confer elite benefits. You still need elite status to benefit from HH upgrades, breakfast etc. to this end the HP discount is bookable on Exec rooms too.

Any company rate offering special local hotel highly discounted rates , ie more than the more usual global 15% off is likely to have your entitlement queried and ID asked for. To be honest my main need is the on day cancelation that corporate HP rate confers

o I am entitled to use HP code (and before that the EDS rate code)
o I have used the HP discount rate in Benelux area, in Amsterdam, Rotterdam, Den Haag, Antwerp and like you said it is a good discounted rate not just the bog standard 15% general corporate discount
+++
o I have been asked to show my Entitlement to the HP discount at checkin in all but Den Haag .. and by time of Den Haag stays I had a Diamond Status
o As a HH diamond elite I also believe I was/am much less likely to be asked than someone like your self with no elite status.