(1)j solution: After passing point D, DM is perpendicular to EF in M and DN is perpendicular to BC in N, respectively.

So angle DMF= angle DME=90 degrees.

Angle DNE= Angle DNC=90 degrees

Because the trapezoid is folded along de.

So CD=FD

EF=EC

Angle definition = angle deviation

Because angle B=90 degrees

So angle B= angle DNC=90 degrees.

So AB parallel DN

Because AD and BC are parallel

So the quadrilateral ABND is a parallelogram.

Because AD=AB=2

So the quadrilateral ABND is a diamond.

So the quadrilateral ABND is a square.

So the angle A=90 degrees.

AD=DN=AB=BN

Because BC=BN+CN=3.

So CN=3-2= 1.

In the right triangle DNC, the angle DNC=90 degrees.

From Pythagorean Theorem:

CD^2=DN^2+CN^2

So FD=CD= root number 5

In the right triangle ADF, the angle α= 90 degrees.

From Pythagorean Theorem:

FD^2=AD^2+AF^2

So AF= 1

Because AB=AF+BF=2

So AF=BF= 1.

So f is the midpoint of AB.

So (1) this conclusion is correct.

(2) Proof: Because angle DME= angle DNE=90 degrees (proved)

Angle Definition = Angle Deviation (Confirmed)

DE=DE

So triangle DME and triangle DNE are congruent (AAS)

So DM=DN

Because AD=DN

So AD=DM

Because angle α = angle DMF=90 degrees

So triangle ADF and triangle MDF are right triangles.

Because DF=DF

So right triangle ADF and right triangle MDF are congruent (HL)

So angle AFD= angle EFD

So FD bisects the AFE angle.

So (2) this conclusion is correct.

(3) Solution: Because the S triangle ADF= 1/2AD*AF.

AD=2 AF= 1

So the s triangle ADF= 1.

Because the s triangle BEF= 1/2*BE*BF.

BE=4/3

BF= 1

So the S triangle BEF=2/3.

So s triangle ADF+S triangle BEF=5/3.

Because angle B=90 degrees

So the BEF triangle is a right triangle.

So from the Pythagorean theorem,

EF^2=BE^2+BF^2

So EF=5/3.

Because the S triangle DEF= 1/2EF*DM.

DM=AD=2

So s triangle DEF=5/3.

So s triangle DEF=S triangle ADF+S triangle BEF

So (3) this conclusion is correct.

(4) solution: because CE=BC-BE

BC=3

BE=4/3

So CE=5/3

Because NE=BN-BE

BN=AD=2

BE=4/3

So NE=2/3

In the right triangle DNE, the angle DNE=90 degrees.

So from the Pythagorean theorem:

DE^2=DN^2+NE^2

So DE=2 times the root number 5/3.

Because CD= root number 5

CE=5/3

So de, CD and be are not equal.

So the conclusion that the triangle DEC is an isosceles triangle is wrong.

To sum up, there are three correct conclusions.

So choose C.