I'm currently doing an autonomous power supply for a small project with an Arduino PRO 5V 16Mhz and sensors. I'm doing it with a solar pannel (10W), a "home made" charge regulator, and a 12V lead acid battery. My charge circuit is quite done.
Depending on the current drawn, the onboard regulator might get hot. To answer your question we really need to know the current drawn by your system (the sensors etc.).
If the project allowed for sleeping time (eg. if you need to sense things every minute) then that would give the regulator time to cool down. It won't get hot with virtually no power going through it.
I'm not drawing much, maximum 150mA under 5V, and the average is probably 100mA. But except for the night, it's drawing permanently.
All my sensors are drawing 25-30mA (5V). My Arduino Pro 20-25mA (5V), and I register values on a SD card (5-10mA) a GPS module 20-25mA (5V) and a RTC (almost nothing).
Or as a last resort you could read the data sheet... and you would have the answers to both questions in under two minutes... about the time since I read your question to now.
The datasheet is attached.