Determining feature importance in decision tree? Generally speaking, if a feature is split early on in a tree does that mean its more important? And so therefore, the root node's feature is the most important feature in the tree?
In addition, if a feature is split often within the tree, does that mean its important as well? Compared to the feature thats split early on, is it more or less important?

These are both broadly true, but not always true.  Of course, it all depends on how you want to measure "important".
Broadly yes, a split higher in the tree was considered alongside other potential splits, and so the fact that this split was made means the tree found it more important than the others (that may get made further down).  And yes, repeatedly splitting on a feature indicates that it is discriminative at several different thresholds and/or in several different slices of the data.
However, there are exceptions.  Consider an XOR style problem: $x_1, x_2$ are i.i.d. centered around zero, and $y=\operatorname{sign}(x_1x_2)$.  There is no good first split: in the population, every split evenly divides the positive and negative class.  But with a finite sample, some split is made, let's say $x_1=0.5$.  After that, splits are much more informative: when $x_1>0.5$, a single split at $x_2=0$ completely separates the data; when $x_1\leq0.5$, likely we'll split also at $x_2=0$ followed by $x_1=0$.  (In practice, some deviation seems to occur; see this notebook I created for this DS.SE question.)  The top split clearly isn't indicative of very much in this case.  Measures of importance like total impurity reduction would be more telling.
Similarly, a very-important binary feature and a somewhat-relevant continuous feature can probably be constructed so that "number of splits" is not the best measure: the binary feature may be the first split, followed by finely chopping up the continuous feature with many splits later.
