The DOT code will have either 8 characters or up to 12 or 13 characters. If looking at the 8 character code the last four will be the date the tire was manufactured in this format: WWYY where WW is the week of the YY year that the tire was made. The other side of the tire will have the longer code. If you are looking at the longer version on the opposite side of the tire it would be the last set of 4 digits for the date of manufacture
For example, 5203 means the tire was made in the 52nd week of 2003 (December, 2003). This tire will be 8 years old and probably needs replacing!!! Another example, 1809, means the tire was made in the 18th week of 2009 (May, 2009) and only about 2 1/2 years old and still good to use as long as the tire has been well taken care of and maintained.
Trailer tires are usually good for 5 to 7 years, but depends on how well they are used and maintained. Factors such as weight of the trailer, how often they are used, psi kept correctly, how they are treated (no high speeds, no curb jumping or cutting, not "left on a turn aspect", etc.), exposed to sun and a few other factors.
If it helps, please post the DOT codes for all your tires (don't forget the spare) and we can help you confirm the age of your tires. If you have questions on tire maintenance, you'll probably need more answers on maintaining the trailer and we can cover those too.
It is not unusual that non-ST tires are put on a trailer. However, this still doesn't mean you can drive or treat them any differently and it's very important that whatever tire you get supports the proper load rating. You might want to post this too so we can tell you what tires you do have. To know what you need (if they are not correct) we need to know the GVWR of the trailer. You either need to locate the sticker on the trailer with this information or weight the trailer (and hope it's below the GVWR).
Sorry I can't help you with the roofing problems. I have done some work on trailer roofs before and it will require scrapping off the gunk very carefully, then using the right stuff made for trailer roofs (don't use silicone). I fear this roof may have been neglected and just covering the problem with new gunk is not a good solution or else you might be covering up any rot under it and this will cause further leakage down the road, or worse yet the gunk was not put at the place the leak was being caused. Remember that the leak in the trailer might be coming from some other arbitrary exterior location and finding them won't be so obvious. Stopping the leak from the exterior is best, then repairing the interior damage is secondary, but necessary to remove mold and damage. Good luck with this one.
That's the spirit, to try and make the best of it.