On the first point:
Days of the week and "n-th of the month" will only line up for consecutive months when a month has a number of days that is a multiple of 7. The only contender for this is February, as it has 28 (excluding a leap year). Other months have a remainder of 31-28=3 or 30-28=2 excess days. When these excess days sum to 7, then it'll be Friday the 13th again.
If we want both February and March to have a Friday 13th, the excess days in the subsequent months don't add up to a multiple of 7 until October, indicating that November would be the next month with a Friday 13th. That only gives three in a calendar year, which feels a little low. But the sequence of months in the end of the year with 31,30,31,30,31 days makes it hard to find a sequence that is a multiple of 7. Having run the numbers in a spreadsheet I whipped up (modulus arithmetic for the win!), I can't find any improvement of 3, even if we allow any 12 consecutive months.
So the best I can find is 3: on the February, March, and November of a given year. (Or February and March, along with the June of the previous year.) Adding a leap year in doesn't increase the total, either.
On the second point:
Using the same spreadsheet, the longest stretch I can find is Friday the 13th of July and Friday 13th of September on the next year, 14 months later.
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