Answer: Based on your description it sounds like you may need to make some adjustments before trying to grow the lawn.
Erosion is caused by water runoff which needs to be redirected or controlled or the problem will get worse over time. Water runs downhill, so you need to look into adjusting or correcting the drainage pattern that affects your yard. Sometimes this type of problem can be easily corrected by lengthening a downspout or adding a swale, but sometimes it is more complex and needs a drain or other extensive measures. I would strongly suggest you consult on site with a professionally trained and degreed landscape architect or engineer with experience in grading and drainage issues and explore your options.
Dogs, wet weather, and lawn are just not a good combination. Running on frozen lawn is also damaging. The dog may need to be given a fenced in dog run area to prevent traffic damage to the lawn. In my experience, a dog who loves to dig may be happy if assigned a "digging area" and then trained not to dig elsewhere. If you can allow the dog's preferred digging spot, this might work for you.
Moles do not usually directly harm plants, they are actually somewhat beneficial in that they aerate the soil and also consume grubs. In general they do not need to be controlled unless they are causing a safety problem. Here is some information about moles you may find helpful.
Lastly, the best time of year for extensive lawn work is actually late summer to early fall so you do have some time. I would recommend you start by running some basic soil tests to check fertility and pH, then adjust those as indicated by the test results. Your local county extension should be able to help you with the testing and interpreting the results as well as with developing a lawn program to renovate or restore your lawn grass. In the meantime they offer some publications on their web site that you may find interesting.
I hope this helps you get started.
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