Charcoal, 94 bytes

ＷＳ⊞υι≔⪫υ¶θＰθ…θ⌕θX≔⟦⟧η≔⁰ζＷ∧ＫＫ∧¬№η⟦ⅈⅉζ⟧∨φ⁻XＫＫ«⊞η⟦ⅈⅉζ⟧¿№ X⊟ＫＤ²✳ζ«Ｍ✳ζ≔⁰φ»≔＆⁷⁻ζ＆³⁻ζ⌕-/|⊟ＫＤ²✳ζζ»Ｔ¹¦¹ 

Try it online! Link is to verbose version of code. Takes input as a rectangular array of newline-terminated strings and outputs X if you reach the starting point. Explanation:

ＷＳ⊞υι 

Input the map.

≔⪫υ¶θＰθ…θ⌕θX 

Write the map to the canvas and start at the X.

≔⟦⟧η 

Prepare to detect an infinite loop.

≔⁰ζ 

Start moving horizontally.

Ｗ∧ＫＫ∧¬№η⟦ⅈⅉζ⟧∨φ⁻XＫＫ« 

Repeat until the cursor moves out of bounds, repeats itself, or moves back on to the X.

⊞η⟦ⅈⅉζ⟧ 

Remember the current position for loop detection.

¿№ X⊟ＫＤ²✳ζ« 

If the next step is out of bounds, empty or the start, then...

Ｍ✳ζ 

... move to that square, and...

≔⁰φ 

... remember that a step was taken.

»≔＆⁷⁻ζ＆³⁻ζ⌕-/|⊟ＫＤ²✳ζζ 

Otherwise, rotate clockwise appropriately to the new direction.

»Ｔ¹¦¹ 

Output the final square.

JavaScript (Node.js), 197 ^{199 205 208} bytes

Gained some bytes by improving solution, but lost some by fixing a bug regarding "bonus case 2"
Gained 8 bytes after analyzing suggestions from ceilingcat !

Interesting challenge!
The input is a single string, with line breaks at the end of each row. Each row must have the same length. Returns true if we can reach the starting point X, and false otherwise.

s=>{i=(d=(p,q)=>p.indexOf(q))(s,"X");l=~d(s,` `);try{(g=(x,y)=>{d(`X `,c=s[z=i+x-y*l]||" ")+1?i=z:0;g([0,x,-y,0,-x|-y,t=x|-y][a=d(`X -|/\\`,c)],[0,y,0,x,x|y,t][a])})(1,0)}catch(e){}return"X"==s[i]} 

Try it online!

This solution uses recursive calls of the g() function and catching exceptions to end an infinite loop (particularly useful for test cases 1 and 4)

Our position is the single int i corresponding to the index in our one-dimensional string. To go up or down, we substract or add the length l of the first row. This works only if all lines have the same length, which is a prerequisite for this code.

In each call we check the cell where we are supposed to go with our current direction (expressed with the x and y parameters of the g() function. These 2 parameters can only have the values [-1, 0, 1]) :

• if it's a space, we change our position to this cell and keep the same direction, then proceed to the next call
• if it's an X or a line break or if our target cell is outside of the string (undefined), we change our position to the index of this cell, and change our direction to [0,0], then proceed to the next call. This allow to "stuck" our cursor on the target cell, and never move while we provoke an infinite loop to end the algorithm
• if we encounter a wall, we don't move but we determine the resulting direction for the next call, and we proceed with this call. For this part i created this tool to help golfing it separately of the whole "mess".
• At the end we check the final position we've reached. If it's X, it's truthy.
• d() function is a shortened indexOf (because we use it more than 2 times)
• in the core mechanism, i used a lot of arrays combined with the already shortened indexOf because i throught it's the shortest way of doing a switch case?

Python3, 484 bytes:

E=enumerate def f(b): r,X,Y,F=[(x,y)for x,r in E(b)for y,c in E(r)if'X'==c],0,1,0 x,y=r.pop(0) while 1: if F==4 or(r and(x,y)==r[0][:2]):return 1 if(len(r)>1 and(x,y,X,Y)in r)or any([(j:=x+X)<0,j>=len(b),(k:=y+Y)<0,k>=len(b[0])]):return B,U,I=b[j][k],X,Y if'|'==B:X,Y=[-1,1][Y>0],0 if'-'==B:X,Y=0,[1,-1][X>0] if'\\'==B:X,Y=[1,-1][max(X,Y)>0],[1,-1][max(X,Y)>0] if'/'==B:X,Y=[-1,1][min(X,Y)<0],[-1,1][min(X,Y)<0] if(X,Y)==(U,I):r+=[(x,y,U,I)];x,y=j,k;F=0 else:F+=1 

Try it online!

Java (JDK), 282 ^{283 287 292} bytes

-2 and -1 bytes thanks to ceilingcat !
-7 bytes with some optimizations

Not bad for Java!

The input is a single string, with line breaks at the end of each row. Each row must have the same length. Returns true if we can reach the starting point X, and false otherwise.

In this solution, we take advantage of the thinness of the boundary between char and int in Java.

s->{int i=s.indexOf(88),l=~s.indexOf(10),x=1,y=0,z,a,c=0,d[],r=2;for(var t=s.toCharArray();r>1;y=d[z+5])x=(d=new int[]{x,-y,0,-x|-y,a=x|-y,y,0,x,x|y,a,r=(z=i+x-y*l)<0|z>=s.length()||(c=t[z])==10|c==40?0:t[i=c>31&c<41?z:i]++>0&c==88|t[i]>95?1:2})[z="-|/\\".indexOf(c)+1];return r>0;} 

Try it online!

This solution transforms the input string into an array of characters, and our cursor "increases" the ASCII values of the characters it passes through.

An infinite path is detected if we pass strictly more than 8 times through the same cell, which would return a false. But if this cell also happens to be our starting point X, then it returns a true.

If we aren't in an infinite loop at all, reaching the X or a \n or having an index outside of the string will return a true or false accordingly.

r is the result and is kind of a ternary boolean: as long as it has the value 2, the algorithm must continue. 0 and 1 are the falsy and truthy values before normalization.

Note:

1. I reused my "wall direction conversion system", and 1-dimensional string indexing from my JavaScript solution.
2. For debugging purpose, we can visualize the path drawn by the cursor (via increase of the ASCII values of the characters) : to do so, move var t=s.toCharArray() with a ; before the for, and add System.out.println(t); just before the return.