Visiting every Circle Line station in a Day

Should be easily done barring major disruption on the line. Flash photography is not permitted on LUL platforms, and if you need to exit each station, a paper OPTC would be best

It should be doable, but photographing the outside of every station will take longer than you think. I recently did this with the Overground circle (Clapham Junction-Willesden Junction-Highbury and Islington-Surrey Quays-Clapham Junction) and it took approximately 8 hours.
The Circle Line covers less distance, but has more stations. The key factor though is waiting for trains, and here the Circle really wins as parts of the overground are only 4tph. For some stations it will be quicker to walk at street level than waiting for trains - Great Portland Street-Euston Road-KXSP and Aldgate-Tower Hill-Monument-Cannon Street-Mansion House (and maybe also to Blackfriars) come to mind as worth investigating. Cannon Street to Monument is only about 2 minutes walk at street level and Tower Hill to Aldgate should be less than 5.

You will probably be OK on Oyster PAYG if you do some walking at street level to ensure that the system recognises you are making separate journeys. It has been known to sometimes get confused by exiting and reentering the same station within a short time if that station is part of an OSI pair. Day travelcards are not available on Oyster.

and because it was designed for normal journeys not for track bashing.

I don't think that leaving stations would be a problem except where there is an OSI. On a couple of occasions I have been caught short on the Met and had to use the facilities at Harrow which are just outside the barriers. I have always been charged for two journeys.

People sometimes visit every pub on the Circle line in a day; I'm sure you'll be able to manage every station...

Better to use Oyster. Paper ODTC may be rejected for re-entry for a while judging from past experience.

All you need to bear in mind with a paper ticket is to avoid getting it bent as that will cause the mag stripe not to work. If you get hold of a ticket wallet from a tube ticket office and just put the paper ticket in that it will do a reasonable job of protecting it - assuming she doesn't put in her back pocket and sit on it If you put another credit card sized bit of card or even an old plastic card in the wallet as well it will help stop the wallet getting bent.

My Overground Circle trip worked as expected on PAYG as well, but on other occasions it hasn't worked. For example I made the following journeys:
1. Debden - Stratford - Canary Wharf DLR
2. Canary Wharf DLR - Bank (exit at Monument)
3. Bank - Angel

However the time I spent outside the system between the journey was less than the OSI time so they were registered as
1. Debden - incomplete
2. incomplete - Angel

Similarly I did Debden - Paddington H&C, saw a friend off on her mainline train, then Paddington Praed Street - Bayswater. The time spent at Paddington was within the OSI limit (just) but the total journey time Debden - Bayswater exceeed that allowed for Z6 to Z1, so it was counted as two incomplete journeys. This was automatically detected after the fact though and my PAYG credit corrected without requiring my intervention.

These were both a few years ago though, and my recent experience is much better. I think I've only had about 3-4 problems in the past couple of years despite all the travelling and odd journeys I do. One of those was due to an error by a DLR PSA, and another was due to all the gates and standalone readers at Woolwich Arsenal NR being set to exit only (with no notice and for no apparent reason as the station was open as usual). One of the others might have been my fault as I'm not certain whether I touched correctly or not.
It seems therefore that Oyster is getting better at dealing with enthusiasts' usage - presumably the years of journey data they now have has helped somewhat with this.

Whether a travelcard or PAYG is cheaper will depend on how many journeys you make and where and what time you are making them. Don't forget you can combine the two - don't get a 7 day Z1-6 travelcard if you're only going beyond Z2 once for example.

As long as you touch in and out correctly, you are unlikely to have any issues with journeys such as you are doing, as even if it does combine a couple of journeys you thought were separate they wont be long enough to come close to maximum journey times (which is what cause the problems I noted above). Altnernating would be a pretty silly method of doing the Circle Line as you'll end up walking some significant stretches and riding some very short ones. On my Overground journey the only stretches I walked were Hampstead Heath to Gospel Oak and Rotherhithe-Canada Water-Surrey Quays, and it all worked fine.

That could be deliberate to avoid encouraging more passengers on to the Vic on that stretch - it's hardly the quietest section of LU's network!