I am staying in a rented house which I cannot sound-proof. I have an acoustic upright piano which I cannot carry to a studio. The noise levels around are average, no heavy traffic, but passing car horns, animals, children playing etc get heard at times. I would like to know, what is the best way to record piano with vocals in this scenario? I am happy with doing live takes with no overdubbing for a start. If I have to choose between a high end dynamic mic and a not-so-high-end condenser mic, what would be better?
Actually you can do soundproofing without damaging the house. Using acoustic panels (expensive but provides the best result) works, however some well placed blankets using string and 3M adhesive hooks in a carpeted room can vastly change the sound of the recording because of reduced reverberation.
Picking a recording time that has the least amount of noise pollution (midnight?) may also help with the external audio source problem.
I would recommend recording vocals and piano separately to achieve the best results. It is difficult if nigh on impossible to properly mic a piano with one mic; let alone a piano and vocalist.
A condenser mic will meet your needs best because even good dynamic mic's can struggle to capture the higher frequencies produced by a piano. Even mid level mics can produce great results if used properly.
The caveat though is placement. You will have to experiment with
placement to find what sounds best. Every audio pro will tell you
that if you have a lousy sound being recorded, fixing it will be difficult at best later.
A condenser can also be used for vocals. Most studio mic's are condensers. It is possible, though unlikely, that the condenser you have in mind is inadequate for recording both audio sources well (provided both are recorded separately, and proper mic placement is used.
To answer your first question there needs to be more detail. Is the piano an upright piano or grand? Do you plan to record only live takes or will you be overdubbing? In answer to your second question, generally condenser mics are a better choice on piano, but again this could use more clarification in regards to specific mics that you're trying to decide between.