Replace missing data

I know this is a duplicate but since I failed to find it with ten minutes of searching:

x = {2, 0, 0, 7, 0, 3, 0, 0, 1};

FoldList[If[#2 == 0, #, #2] &, x]

{2, 2, 2, 7, 7, 3, 3, 3, 1}

To handle leading zeros:

x = {0, 7, 0, 3, 0, 0, 1};

FoldList[If[#2 == 0, #, #2] &, SelectFirst[x, # != 0 &], Rest @ x]

{7, 7, 7, 3, 3, 3, 1}

The case without leading zeroes can also be handled nicely using Split, borrowed from Leonid's method for Fill out blanks with a upcoming number in a list?:

x = {2, 0, 0, 7, 0, 3, 0, 0, 1};

Join @@ Accumulate /@ Split[x, #2 == 0 &]

{2, 2, 2, 7, 7, 3, 3, 3, 1}

The leading zeros can be handled by prepending the first non-zero value to the list, then applying Rest at the end.

For Q2 I think all you need is to convert either method to a function and map it:

fn[x_] := FoldList[If[#2 == 0, #, #2] &, SelectFirst[x, # != 0 &], Rest @ x]

fn /@ {{2, 0, 0, 7, 0, 3, 0, 0, 1}, {0, 1, 2, 7, 0, 3, 0, 0, 1}}

{{2, 2, 2, 7, 7, 3, 3, 3, 1}, {1, 1, 2, 7, 7, 3, 3, 3, 1}}

TemporalData + MissingDataMethod

ClearAll[interpolateMissing]
interpolateMissing = Quiet @ TemporalData[# /. 0 -> Missing[], Automatic, 
  MissingDataMethod -> {"Interpolation", InterpolationOrder -> 0}]["States"] &;

Examples:

interpolateMissing  @ {1, 0, 2, 0, 3}

{{1, 1, 2, 2, 3}}

interpolateMissing  @ {0, 7, 0, 3, 0, 0, 1}

{{7, 7, 7, 3, 3, 3, 1}}

interpolateMissing  @ {0, 0, 0, 0, 7, 0, 3, 0, 0, 1}

{{7, 7, 7, 7, 7, 7, 3, 3, 3, 1}}

At the price of extra pair braces in the output when the input is a list, we get the second requirement

How can I expand it to go over matrices of equal-length-vectors?

for free, since interpolateMissing works as is when the input is an array:

interpolateMissing  @ {{1, 0, 2, 0, 3, 0, 1}, {0, 7, 0, 3, 0, 0, 1}}

{{1, 1, 2, 2, 3, 3, 1}, {7, 7, 7, 3, 3, 3, 1}}

Note: Quiet is used to suppress the following warning:

InterpolatingFunction::dmval: Input value {0} lies outside the range of data in the interpolating function. Extrapolation will be used. >>

By the way, Extrapolation is exactly what we need to handle cases with leading zeros according to requirement (3).