Heathrow Departure Tax?

There is no specific "Heathrow" departure tax, but rather a UK tax. If you look at the details of any airline ticket that you purchase, you will see a long list of taxes and fees. Taxes are included in the total price of your airline ticket. If you are buying a ticket, just compare the total price. The taxes and fees matter when people use Frequent Flyer miles. The free ticket is not really free. You have to pay the taxes for the "free" ticket that you get. The UK taxes are higher than in other countries, so if you are using frequent flyer miles and want the ticket to be as cheap as possible, the UK is usually not the best country to fly to and from.

APD is only paid on flights departing the UK, not arriving at it, and connections via the UK are only charged if there is 24 hours between the flights. So if you are arranging an open jaw, arrive in the UK and depart from elsewhere. Note that the cost of the APD can be offset by the UK having the cheapest available fare, especially in economy class.

I've flown through LHR for years. The only "departure" tax I am aware of is if you upgrade your seat (from coach to business class). There is a "luxury" tax of some sort that gets applied for the outbound leg from the UK on an upgraded flight (the presumption is that you are using miles to upgrade). I have upgraded several times using airline miles, and there is always a tax on the upgrade for the leg originating in the UK. I think the last time I paid it, the tax was $90, flying American. Perhaps this is what you are referring to?

There is a 'reduced' rate of APD charged to economy passengers, and a standard rate paid in higher classes. If you decide to upgrade your class from economy, you lose the 50% discount.