Welcome to the forum.
The fare you're charged when you use Oyster or contactless is the PAYG fare, which you can see on the TfL single fare finder. These PAYG fares are sometimes cheaper than conventional singles/returns, and sometimes more expensive - it depends on the precise circumstances.
Heading
into London, PAYG off-peak fares are charged if you touch in before 06:30 or after 09:30 - including during the 'evening peak'. Heading
out of London the same times apply in the morning, but you are charged peak fares if you touch in between 16:00 and 19:00 in the evening. So it is the time that you touch in at the
start of your journey that matters, and you can pay off-peak fares by touching in before 06:30.
The fares you're seeing on the c2c website are conventional singles/returns - which you'd usually have as a paper ticket, although in some cases you may be able to load them to a smartcard instead. For this particular journey they are slightly more expensive than the PAYG option. The restrictions are also different - the Anytime paper fares apply for trains departing between 04:30 and 09:29 in the morning, and 16:00 and 18:59 in the evening, in
both directions.
The above timings are based on the
scheduled departure time of the train you want to catch, rather than when you touch in. When there's disruption, this can make a bit of a difference - e.g. in the evening peak, you could take a late-running 15:50 train with a paper Off-Peak ticket but you'd be charged the peak PAYG fare. Anytime paper fares apply for all trains departing between 04:30 and 09:29, so there's no "early morning" off-peak option like you get with PAYG fares.
So in answer to your questions, if you switch from Oyster to contactless you would still be using PAYG so you would pay the fare shown on the TfL single fare finder (which includes the 'early morning' pre-06:30 off-peak option). If you travel before 06:30 and went for a paper ticket instead, you would have to pay for an Anytime ticket, since there is no 'early morning' Off-Peak paper option.
If you commute 5 days a week, it might be worth considering a paper/smartcard season ticket from Grays to Fenchurch Street (or West Ham, depending on where you change to the Underground). This would cost £75.40 or £63.00 a week, respectively - the equivalent of £15.08 or £12.60 a day. There are discounts for getting a longer season ticket (monthly, annual, or any duration in between).
Alternatively you could also consider getting a Flexi Season for the rail part of the journey - this is in fact slightly cheaper than a weekly season ticket (£14.83 or £12.55 a day, respectively), but is more flexible as you don't have to commute in 5 days a week. The only disadvantage is that you wouldn't get free travel to London on the weekend. Each Flexi Season is valid for any chosen 8 days within a 28 day period, so you only need to commute an average of 2 days a week to use up all the days. A Flexi Season can only be obtained on a smartcard, or on an app - you need to activate a day each time you want to travel.
If you qualify for railcard note these can also only be added to Oyster. This excludes the Network Railcard but does include the Annual Gold Card (which anyone can buy for around £200 with a cheap Hatton-Lapworth season ticket), which gives a discount on travel after 0930.
For Oyster, Railcard discounts on apply to off-peak single fares whenever they are charged (including in the pre-06:30 period). You are perhaps thinking of the off-peak daily caps, which only apply to travel from 09:30 onwards.
This has advantages (it means the OP could get a Railcard discount on their pre-06:30 journeys) but also disadvantages (they would not get any discount on Oyster fares in the evening peak).
However, this does provide the OP with an opportunity. They could buy a cheap season ticket qualifying for an Annual Gold Card (as per the above suggestion) and apply this to their Oyster card. They would then use the Oyster card in the morning, touching in before 06:30 to pay the Railcard-discounted Off-Peak fare (£6.25) on their way into work.
On their way home (presuming they'll be starting their return journey between 16:00-19:00) they could then buy a
paper Anytime ticket discounted with the Annual Gold Card, costing £7.10.
This would mean their commute costs £13.35 per day, which is quite a lot cheaper than any of the other options I can readily find.
