Annoyingly, Heathrow isn't a "proper" railway station. You might think it is, and it certainly looks at first glance like one, but apparently it isn't... The practical upshot is that you can't collect normal train tickets booked online from Heathrow, and they won't sell you train tickets for anywhere further west than Reading, or anywhere further east than London Paddington. (However, if you turn up at the ticket office with an online booking code and a confused look, along with a printout of the ticket bought, they'll often give you a hand written ticket to get to Paddington to collect, but I'm not sure it can be guaranteed on)

The peak/off-peak thing varies depending on the train company that prices the journey (which might not be the one you travel on...)

Southend Victoria to London is £14.90 off-peak, £15.70 peak, but there don't seem to be very many peak time trains in the afternoon - most seem to allow off-peak ticekts. Southend Central is cheaper at £11, and there's only peak time tickets so for that £11 you can travel any time of the day. There's no multi-day returns on London to Southend Central, so it's just £11 each way / £22 total. A multi-day (off peak return) from London to Southend Victoria is £23.70, and there's only a few trains you can't get, but you're probably better off into Central for £22.

Prices for through tickets from Heathrow to Southend look crazy, so you're probably best off buying a tube ticket (assuming no Oyster?), take the tube into London, then buy an £11 single at Fenchurch Street