E1 elimination and alkyne formation Are E1 elimination reactions possible from a trans dihalide alkane (like 1,2-dibromoethane) all the way to the alkyne (like acetylene)? Or is only E2 possible from the alkene to alkyne? What is the rate of reaction?

My thoughts are that E1 is a possible reaction to form the alkyne, with E2 taking place if there are any anti peri-planar hydrogens and a strong base. So, another question might be that which of these has the faster rate of reaction - an E1 reaction or an E2 reaction with anti periplanar hydrogens that are only available through conformational energies and torsional strain?
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.