The reason that the date will not stay is that the "date battery is dead". Please see pages 114 & 115 of the Advanced manual to change it. You can download a PDF copy at the link below.
The second issue may be a little more tricky. First check to see fi the lens covers are opening completely when you power the camera on. If not, you will need to exercise them to free up their travel.
If the covers open correctly, then, unfortunately what you have described indicates that there is a problem with the ccd (the light sensitive chip that produces the image)or the connection to it, and while your camera is not on the current list of Canon cameras for possible free repair, the information at the links below do fully explain the problem and tells how Canon handled it. Since more and more newer models have the identical problem the info may give you some leverage with Canon. You are the 5th person in a one month period, with a Canon camera of similar vintage that I have advised of this problem. Click on or copy and paste the link into your browserhttp://www.usa.canon.com/cusa/support/consumer/digital_cameras/powershot_a_series/powershot_a75?pageKeyCode=prdAdvDetail&docId=0901e024801e74fehttp://www.imaging-resource.com/badccds.htmlIf a conventional repair is the option you are left with, considering the age and present value of this camera, paying the cost of having it repaired , approximately $75 - $90 (if the parts can be obtained) really is probably not very cost effective. The standard "rule of thumb" is if the repair cost is greater than 50% of the value of the camera, it is not worth it. A used or refurbished Powershot A530 can be purchased for about $15-$40
Please keep in mind that my diagnosis & solutions provided are directly dependent on the accuracy of your description of the problem. As with any "do it yourself" fixes, success is a "team effort", since I can't see or touch the camera, and relies on the customer's manual dexterity and ability to follow the instructions well.If you require additional information, please use the Reply to Expert Tab to contact me. Click the ratings only after our exchange is complete and you are satisfied.If, and only if my answer to your question is helpful and you are Satisfied with it, please leave Positive Feedback.
Please double check the seating of the "date battery". Make sure that there is not a clear plastic insulation piece over the negative surface. Even if the other batteries are good the date will not stay without a good CR1220.