Yes.. Sure ..You can use the same machine...Procedure is same for both..only difference is in the loss calculations for(cable attenuation per Km(dB)..For 1310nm it is 0.38 and 1550 it is 0.25) ..that's all ..go ahead ..
No, you cannot use a multimode otdr for singlemode testing. The results will be totally inaccurate. First of all, multimode fiber has a different core diameter; either 50um or 62.5 while singlemode fiber is 8 - 10um core. You probably won't even get a descent launch into the fiber. Additionally, the losses/kilometer and designed wavelengths are very different and so will be the index of refraction. All of this will provide no worthwhile information. Any splices, or other anomalies will show up at an erroneous distance if you get anything at all. If your OTDR is modular, you may be able to rent a singlemode module. It not, just rent a singlemode OTDR and get on with it.
PS: Be sure you get the right connector adapters as well.