Why wouldn't anyone count the money?
They count it in shops when you pay for goods.
No excuse needed.

I always count the money at the start of a booking. If any looks or feels off, i will even blacklight it. No excuses needed, as it is an agreed transaction at the end of the day. Only clients that tend to have an issue are those that have short changed me and know i will notice. Once i start counting, this typically leads them to start grumbling and then hand me the rest of the fee. Even with long term regulars, i will count. It just becomes routine and they never mind.

Yes, I count when I'm a new customer, but when I'm regular I don't count in front of them anymore.

I always count in front of them but discreetly. I take the amount and spread it between my fingers to see if everything is correct and while I am going to put it away I touch the texture well to make sure they are not fake and check with a counterfeit detector if I have any doubts.

Yes, every single time. Regular or not. I've been stung a few times as a newbie, and it was mostly by people I'd seen before (because they think you'll let your guard down). Stay vigilant! You can always say it's for their protection too against any disputes/checking they haven't accidentally overpaid.

It's best to get the money part of the business out of the way as near to the start of the booking as possible in my opinion, as long as you don't seem like you're trying to mug him 

I don’t see the issue. If the money is all present and correct it shouldn’t be an issue. They shouldn’t worry.
Better to count it there in front of them rather than doing it in the next room and then returning to say it’s not all there.
There’s nothing rude or unsavoury about it. People in shops count money in front of you, why is this different?