E1 elimination and alkyne formation
Yes, in your example, it is possible to form acetylene. E1 mechanism does not depend on whether the leaving group is anti to $\ce{H}$ or not.

All vicinal halides will not give alkynes, as E1 mechanism involves a carbocation intermediate which can rearrange to the most stable position. Example: In 4-methyl-2,3-dibromopentane, the product will be a conjugated alkene not alkyne.

Rate of E1 elimination depends on the stability of the carbocation whereas rate of E2 mechanism does not. Since in your example, the carbocation formed is a 1 degree, it is not very stable so the rate will be higher for E2 mechanism.