Oyster Perpetual Day Date keeps stopping

You can, but putting in back on and not cross threaded is a lot more difficult, plus it has a sealing gasket that need replacing. If you want to play with a $10,000 watch, fine. :>)

Hold the watch firmly, turn the ball to the left until it starts to unscrew. Keep going until you can grip the back of the watch with your fingers and turn it some more. Pull the back off.

How to Open a Rolex Watch: 10 Steps (with Pictures) - wikiHow

https://www.wikihow.com > ... > Jewelry > Watches

They don't have battery's ,it is a analogue watch, winding .

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Try this link to get more details. Gopakumar Gopalan Google

The Rolex time change is done after three clicks when you pull the winder. If the winder can actually be pulled out of the stem (separate) the damage is serious.
Sorry. Only an authorized dealer can get the parts for it to be repaired.

this link will help you
Identify Counterfeit Fake Rolex Watches Rolex or Replica Don get Screwed

Hi , I am sorry to say that a rolex requires servicing on a regular basis . Every 3 - 5 years would be advised . If you consider that your watch is a delicate machine that may be working 24 hours a day for years then its not unreasonable that it needs servicing. Over time the fine oils dry and degrade , they need replacing , and parts can become damaged .
As to cost , this is a skilled job . To service a rolex in the correct way requires not only skill ,but the correct tooling and equipment . Also the materials used must be genuine rolex , this is costly . Your accredited service centre will then provide a 2 year guarantee .
All in all the price of $600.00 reflects the labour of a skilled man and his investment in gaining rolex status .
There will be cheaper alternatives but beware as they will probably not be rolex trained and wont be using original materials .

the shaft on one side of the balance wheel as broken and needs to go to jewelers for repair.

https://youtu.be/utu7EA8Uq88
Watch the above

Reset the time, day and date manually from scratch. Check for another 24 hours. If the problem persists that means there is interference between day and date wheels and in this case you will need watchmakers help, as to correct such a fault the watch needs to be disassembled almost completely.

There isn;t any downloadable manuals for Rolex watches, but operation of Rolex Day-Date is very simple:
1) when you unscrew the crown, it is in winding position,
2) when you pull out the crown to the first click, it is in day and date setting position (depending wich way you turn the crown).
NOTE: to adjust the day or date, previously you have to advance hands forward, passing 12 o'clock (midnight) mark and stop advancing at 4.00 a.m. using the crown in position as described in chapter 3. When done - push the crown fully in, then pull out to the first click and only then adjust the day or date.
3) when you pull the crown further out to the second click, it is in time setting position.
To find the daytime or night time on the watch: pull out crown to the second click and advance watch hands forward till you pass the 12 mark on dial. If the date changes as you pass it - it is the night time (midnight). If the date do not change as you pass it - it is the daytime (noon).
After all settings being done, allways screw the crown back down as it protects the watch from water and moisture.
Never use watch in sauna- it's not designed for that.
Do not forget to rate, please.