Thank you, Gustavo!
(a) First, there is a problem with government funding for schools.
Your variant is more logical because mine might sound as if government funding itself is a problem. I would change it to "the problem of inadequate school funding".
Yes, it can. This must be actually the only case I know of where "there" can anticipate a defined noun phrase, and this only occurs with enumerations.
Does it mean that if I change "first" to "for example" I should use "a problem of"?
Here's a passage from The New York Times:
"While food safety issues should be addressed, some scientists say, the bigger concern is the environmental threats posed by genetically modified animals like the salmon.
A recent study showed, for example, that populations of wild fish could, in theory, be wiped out by mating with certain kinds of genetically engineered fish, should they escape.
In addition, there is the possibility of unpredictable environmental disruptions, like those that occur when non-native species invade ecosystems, as the zebra mussels have the Hudson River."
There is no enumeration there. Does it look correct to you?