I agree that both of these questions are ambiguous. It's not just that they aren't fully unambiguous themselves. There are many people out there who ask sloppy questions, which sound like they're asking one thing, but mean something slightly different. These questions are not clearly different from a sloppy question.
Question 1: The clock circuit (oscillator circuit) and reset circuit are internal circuits.
Scenario A
- Student: They're internal circuits. See, the oscillator circuit is the amplifier inside the chip, and the reset circuit is this beast:

- Professor: No, they're external. The PIC16F877A has no internal oscillator, and when I said 'reset circuit', I meant MCLR circuit, which is external:

Scenario B:
- Student: They're external circuits.
- Professor: No, they're internal circuits.
Both of these scenarios, while not equally likely, are both totally plausible. The professor does not make it clear that he's not asking a sloppy question.
Question 2: What is the roll over value of Timer1 when operating in normal mode?
My problem with this question is that it's badly formed in the first place, meaning the answer he's looking for isn't clear. Multiple choice questions like this are annoying because you don't get a chance to explain your answer. This question could be answers in three ways (all of which would demonstrate that the student fully understood the timer):
Answer 1: None of the above. The timer rolls over between 0xFFFF and 0x10000. It's kind of meaningless to talk about the roll over value.
Answer 2: 0xFFFF is the rollover value because it's the highest value the 16-bit timer can store. After this, it rolls over.
Answer 3: 0x10000. Because this is the value that actually triggers the rollover.
0x0000 - not rolled over
0x0001 - not rolled over
...
0xFFFD - not rolled over
0xFFFE - not rolled over
0xFFFF - roll over happens after this
0x10000 - roll over has happened! -> 0x0000
I can hear someone shouting that a 16-bit timer can't hold 0x10000 because it's a 17-bit value. What they don't realize is that the timer is a 17-bit counter. The 17th bit is the TMR1IF bit, which is sticky.
bit: IFEDCBA9876543210 (bit I is TMR1IF)
00000000000000000 - not rolled over
00000000000000001 - not rolled over
00000000000000010 - not rolled over
00000000000000011 - not rolled over
...
01111111111111100 - not rolled over
01111111111111101 - not rolled over
01111111111111110 - not rolled over
01111111111111111 - rollover happens after this
10000000000000000 - rollover has happened!
This is how I would have asked the questions:
Question 1: (Actually, I'm not sure what is the point of this question. The oscillator and reset circuits are implemented partly inside and partly outside the device.)
Question 2: When Timer 1 is operating in normal mode, what is its last value before it rolls over.
P.S. sorry if there are any more typos or nonsensical sentences in this answer. Apparently I'm still suffering the effects of the general anesthetic this morning.
Best Answer
Because embedded systems often need to generate time intervals with many different periods or frequencies, for things like:
Having dedicated timers for each of these means that the operation is more autonomous, and fewer CPU cycles need to be spent on keeping them going.