So first question, which of these is better practice and what exactly is the first one doing?

df['Action'] = df['param1'].map(my_function)

is much more idiomatic, faster (vectorized) and more reliable.

2nd and final question. This is the more important of the two. Map, apply, applymap - not sure which to use here. The first commented out line of code does NOT work, while the second gives me exactly what I want.

Pandas does NOT have DataFrame.map() - only Series.map(), so if you need to access multiple columns in your mapping function - you can use DataFrame.apply().

Demo:

df['New_Col'] = df.apply(lamba x: my_function(x.param1,
                                              x.param1.shift(1),
                                              x.param2.shift(1),
                         axis=1) 

or just:

df['New_Col'] = my_function(df.param1, df.param1.shift(1), df.param2.shift(1))