I took a look at Schinzel and Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arith IV (1958) 185-208, reprinted in Volume 2 of Schinzel's Selecta, pages 1113-1133. In the Selecta, there is a commentary by Jerzy Kaczorowski, who mentions "the G H Hardy and J E Littlewood conjecture implicitly formulated in [33] that $\pi(x+y)\le\pi(x)+\pi(y)$ for $x,y\ge2$." [33] is Partitio Numerorum III. Schinzel and Sierpinski (page 1127 of the Selecta) define $\rho(x)=\limsup_{y\to\infty}[\pi(y+x)-\pi(y)]$, and point to that H-L paper, pp 52-68. They then write (page 1131), "$\bf C_{12.2}.$ L'hypothese de Hardy et Littlewood suivant laquelle $\rho(x)\le\pi(x)$ pour $x$ naturels $\gt1$ equivaut a l'inegalite $\pi(x+y)\le\pi(x)+\pi(y)$ pour $x\gt1,y\gt1$." It should be said that the proof that the first inequality implies the second relies on Hypothesis H, which essentially says that if there is no simple reason why a bunch of polynomials can't all be prime, then they are, infinitely often.
Schinzel and Sierpinski express no opinion as to any degree of belief in the conjecture under discussion.
I don't suppose this actually answers any of the questions, although Kaczorowski's use of the word "implicitly" may be significant.
I think that this question is somewhat similar to:
"How does a botanist decide which plants to look at?"
Well... a botanist always looks at lots of plants.
In the early phases of his/her career, a botanist will spend a lot a time studying well-known plants. Later on, a botanist will have learned to quickly recognise the commonly occurring plants so as to be able to quickly ignore them. He/she will then be able to focus their energy on rarer plants, or plants whose morphology is particularly interesting.
Even later, a botanist might have the opportunity to participate in expeditions to various remote parts of the planet, in order to search for plants that are new to science.
Now:
"How does a mathematician decide which problems to think about?"
Well... a mathematician thinks about a lot of stuff.
In the early phases of his/her career, a mathematician will spend a lot a time studying well-known constructions. Later on, a mathematician will have learned to recognise those problems that can be treated efficiently with well-known tools from those which are likely to be too hard. With the help of an adviser, he/she will then be able to focus their energy on problems that are at the right level of difficulty. Here, the meaning of "the right level of difficulty" always depends a lot on the expertise and background.
Experienced mathematicians have had the possibility to accumulate, throughout their interactions with colleagues, and by their own personal attempts at solving problems, a little collection of problems that are specialized enough so that no-one has really thought about them yet, and not too difficult for a graduate student. That's how graduate students often get their problems to work on.
The mathematicians who are no-longer graduate students have to constantly try to solve new problems. Their ability to find interesting problems (and to solve them) defines how good they are at their job.
Best Answer
Existence or nonexistence of odd perfect numbers.
Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html Nichomachus also asked about infinitude of perfect numbers.
(Goes back at least to Descartes 1638 http://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)