I'll try to re-describe the same ideas in a different way. This isn't meant to be a quick answer to the question; rather, this is meant to be a resource to help build some intuition.
In this answer, the word "frame" is not used. That's because "frame" might carry connotations of something rigid, something defined by "axes." This answer is expressed using the more general concept of a coordinate system, which doesn't rely on anything like axes or straight lines.
Coordinates are arbitrary labels for the points in spacetime. They should assign a unique 4-tuple of numbers $(w,x,y,z)$ to each point, and they should do this in a smooth way, but otherwise they are arbitrary. Any worldline (curve in spacetime) can be described by giving the four coordinates as functions of some other parameter $\lambda$ that runs along the worldline. Examples will be shown below, after the general principles.
Mathematically, coordinate systems and worldlines are defined without the help of any geometric concepts like time, distance, timelike, or spacelike, and without the help of any dynamic concepts like free-fall. Geometry (including time) and free-fall are both defined instead by the metric. A convenient way to specify the metric is by specifying the line element. The line element takes any $\lambda$-parameterized worldline as input and returns a single function $G(\lambda)$ as output. In special relativity, the line element can be expressed as
$$
G(\lambda) = \dot w^2 - (\dot x^2+\dot y^2+\dot z^2)
\tag{1}
$$
where a dot denotes a derivative with respect to the parameter $\lambda$ along the given worldline. The worldline is called
timelike wherever $G(\lambda)>0$,
spacelike wherever $G(\lambda)<0$,
lightlike wherever $G(\lambda)=0$.
A worldline is called causal if it is either timelike or lightlike. The causality principle says that only a causal worldline can represent the history of a physical object. Proper time is defined only along such a worldline. Given any causal worldline, its proper time $\tau(\lambda)$ is defined by the condition
$$
\dot\tau^2 = G(\lambda) \geq 0.
\tag{2}
$$
This tells us how the proper time $\tau$ progresses along the worldline as a function of the parameter $\lambda$.
In hindsight, now that the line element (1) has been specified, we see that a wordline cannot be timelike unless $w$ changes monotonically along the worldline. In this sense, we can think of $w$ as a "timelike" coordinate — but it is still just a coordinate. Proper time is given by equation (2), and this is what an object actually experiences as time. Proper time is specific to the given worldline, and it is invariant under changes of the coordinate system.
If the quantity (1) is negative, then we have a spacelike worldline. Proper time is undefined along such a worldline. Physical objects, including clocks, cannot move according to such a worldline, so we shouldn't expect to have any invariant notion of the progression of time along such a worldline. What we have instead for such a worldline is proper distance $\ell$, given by the condition
$$
\dot\ell^2 = -G(\lambda) > 0.
\tag{3}
$$
Two points in spacetime are said to be "timelike separated" if they can be connected to each other by some timelike worldline, and they are said to be "spacelike separated" if they cannot be connected to each other by any causal worldline. The concept of "spacelike separated" events is an extension of the concept of "simultaneous" events. Spacelike-separated events cannot be time-ordered in any invariant way.
By the way, even if two points are timelike separated (which means that one of the points is unambiguously in the future of the other), they can still be connectd to each other by a spacelike worldline. The following pair of examples illustrates this.
Example 1
Choose constants $A,B,C$ and consider the worldline given by
$$
w(\lambda)=A\lambda
\hskip1cm
x(\lambda)=B\lambda+C
\hskip1cm
y(\lambda)=0
\hskip1cm
z(\lambda)=0.
\tag{4}
$$
Then
$$
\dot w = A
\hskip1cm
\dot x = B
\hskip1cm
\dot y = 0
\hskip1cm
\dot z = 0,
\tag{5}
$$
so $G(\lambda)=A^2-B^2$, which is independent of $\lambda$ in this simple example. This worldline is:
timelike if $A^2>B^2$, and then equation (2) gives $\tau(\lambda) = \sqrt{A^2-B^2}\,\lambda$ for the proper time along this worldline.
spacelike if $A^2<B^2$, and then equation (3) gives $\ell(\lambda) = \sqrt{B^2-A^2}\,\lambda$ for the proper distance along this worldline.
lightlike if $A^2=B^2$, and then the proper time and proper distance are both zero along this worldline.
For the special metric defined by (1), a timelike (or lightlike) worldline correpsonds to free-fall if and only if the derivatives $(\dot w,\dot x,\dot y,\dot z)$ are all proportional to each other. In particular:
The worldline defined by (4) represents free-fall if $A^2\geq B^2$.
If $A^2<B^2$, then it does not represent any physically possible motion.
Example 2
Consider the worldline defined by
\begin{align}
w(\lambda) &= \lambda + \lambda^3 \\
x(\lambda) &= \cos(\beta\lambda + \beta\lambda^3) \\
y(\lambda) &= \sin(\beta\lambda + \beta\lambda^3) \\
z(\lambda) &= 0.
\tag{6}
\end{align}
where $\beta$ is a constant. For each value of $\lambda$, these equations specify the coordinates of one point in the four-dimensional manifold, so they define a worldline. Plug (6) into (1) to get
$$
G(\lambda)=(1-\beta^2)(1+3\lambda^2)^2.
\tag{7}
$$
If $\beta^2<1$, then this worldline is timelike, and then equation (2) says that its proper time is given by
$$
\tau(\lambda)=\sqrt{1-\beta^2}\,(\lambda+\lambda^3).
\tag{8}
$$
This tells us how the proper time $\tau$ progresses along the worldline as a function of the parameter $\lambda$. Equation (8) is independent of the coordinates, as it should be; proper time is invariant under coordinate transformations.
If $\beta^2>1$, then this worldline is spacelike. Notice, though, that this spacelike worldline passes through all of the points $(w,x,y,z)=(2\pi\,n/\beta,1,0,0)$ for all integers $n$, and these points are also all contained in the timelike worldline (4) with $A=1$, $B=0$, and $C=1$. (The parameters $\lambda$ of the two worldlines are not the same; the symbol $\lambda$ was recycled.) This shows that the worldline (6) with $\beta^2>1$ is an example of a spacelike worldline that connects some timelike-separated points.
