The best time to grow grass is late summer through mid October. It would be okay to spot seed now. Keep watered until the grass germinates and is up and growing. The second best time is in the spring March through early April.
The optimum pH for turf growth is 6.0 to 6.8 You pH sounds fine.
Apply lime only according to soil test results. A soil test will give pH, liming, and nutrient deficiencies. https://extension.umd.edu/hgic/topics/soil-testing
If soil test results show you need to apply lime, the ideal time is
in the fall. https://extension.umd.edu/hgic/topics/liming-lawn
If I use fast acting lime granuals do I need to wet grass after or before?
We are not aware of a need to wet grass before or after applying fast-acting lime. However, read the label of the product you use.