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Three of four speakers pronounce 二月 as LHH, and one (from the US) pronounces it as HLL.
As a rule of thumb 月 (gatsu) and 日 (nichi) are 尾高 when they follow accented number, and 頭高 when they follow nonaccented number (3,5,9) (9 is considered nonaccented only as く, ...
Regarding counters there are no rules in general. But some kinds of counters are more or less predictable.
One mora counters
One mora counters are preaccented in general: 個,里,羽 etc (一個 [Ikko],一羽 [ICHIwa]...). One useful exception is 度 [do] when used as 'times', though it's also preaccented when used as 'degrees'
Three or more mora counters
Three or more ...