I'll try to answer the questions posed in the edit. To start with, I'll talk through the various methods for performing braids using T-junctions. Then I'll discuss the last question, about projective measurements.
Braiding with T-Junctions
To start with the proposal of Figure 3 in the review paper you cited, it might be worth going to the original paper (https://www.nature.com/articles/nphys1915) for a more in-depth explanation, particularly the supplemental information.
Basically, unpaired Majoranas in nanowires are pinned to the boundary between topological and non-topological regions of the superconducting wires. Whether or not a region is in the topological phase is controlled by electrical gates which can change the chemical potential of the region. So by tuning these gates, you can extend or shrink the length of a topological region in the wire, thus moving the Majoranas at its boundary.
In terms of how to move Majoranas past a T-junction, have a look at Figure 2 in the supplementary info of the paper I linked. We can think of a T-junction as being the spot where three different Majorana-carrying nanowires meet. Each of these wires can host Majoranas at its ends. However, there are tunnel-couplings set up between the endpoints of the three wires located at the junction, meaning that only one zero-energy Majorana can sit at that junction, while the other two Majoranas pair up into a finite energy fermion. This basically means that if an even number nanowires around the junction are in the topological phase, there is no unpaired Majorana located at the junction (both Majoranas located at the junction are paired up into a finite energy fermion), while if there are an odd number of topological wires around the junction, there is an unpaired Majorana located there. This effectively means that you can move Majoranas through the junction onto any of its branches by tuning the electrical gates.
So to braid two Majoranas, you move them around the T-junction according to one of the patterns shown in Figure 3 of the paper.
There are more-complex "flux-controlled" braiding proposals, like that of this paper (https://arxiv.org/abs/1111.6001). The idea with these proposals is that you have three topological nanowires meeting at a T-junction, which results in a single zero-energy Majorana located at the junction $\gamma_0$, along with three other Majoranas located at the other ends of the three wires, $\gamma_1$, $\gamma_2$, $\gamma_3$. The effective low-energy Hamiltonian of this system includes three terms, $\Delta_k i \gamma_0\gamma_k$ for $k=1,2,3$, with strengths $\Delta_k$ that can be tuned.
By tuning these three $\Delta_k$'s adiabatically you can braid any two of the Majoranas $\gamma_1$, $\gamma_2$, $\gamma_3$. This procedure is effectively an adiabatic version of a measurement-only protocol which I'll explain below.
Measurement-Only Braiding
These protocols involve a series of bilinear measurements - measurements of operators of the form $i\gamma_j\gamma_k$. For example see this paper (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.010501) on measurement-only topological computation with arbitrary anyons, and this paper (https://arxiv.org/abs/1909.03002) for a more recent Majorana-specific discussion.
The idea is that you need two ancillary Majoranas, which we will call $\gamma_a$ and $\gamma_b$, in a known parity state $i\gamma_a\gamma_b = 1$, on top of the two Majoranas you wish to braid, $\gamma_1$ and $\gamma_2$. Suppose these Majoranas form a topological qubit with two others, $\gamma_3$ and $\gamma_4$, which is in the state $i\gamma_1\gamma_3 = +1$.
If we make a measurement of the operator $i\gamma_1\gamma_a$ and determine the result to be $+1$, we have effectively transferred the "ancilla state" to now lie between anyons $a$ and $1$. But the original logical state of the qubit stored between Majoranas 1-4 is not destroyed by this measurement since you cannot retrieve any information about the original state from the measurement. Hence, the logical state now must be stored in the anyons $\gamma_b, \gamma_2, \gamma_3,\gamma_4$.
You then similarly measure the operator $i\gamma_2\gamma_a$, and if the result is $+1$, the logical state is stored in Majoranas $\gamma_b,\gamma_1,\gamma_3,\gamma_4$ while the ancilla state is in $\gamma_2$ and $\gamma_a$. Finally we measure $i\gamma_a\gamma_b$ again. And if the result is once again $+1$ the logical state is now stored between $\gamma_2,\gamma_1,\gamma_3,\gamma_4$ and the ancilla state is back to where it started.
So how is the logical qubit affected? It started in the state $i\gamma_1\gamma_3 = +1$, then after the first measurement the logical Majoranas are in an eigenstate of $i\gamma_b\gamma_3$. This operator commutes with the second measurement but after the third measurement, the logical Majoranas are in an eigenstate of $i\gamma_2\gamma_3$. Hence the series of measurements have resulted in the transformation $\gamma_1 \rightarrow \pm \gamma_2$. Similarly you can show that $\gamma_2\rightarrow \mp \gamma_1$ (I haven't worked out the signs but one can do so).
These measurements have to produce the result $+1$ for the procedure to work as specified, so you more or less have to repeat the protocol until you get the desired measurement outcomes.
