Depends on what version you buy.
Lesser Rod of Giant Summoning can do up to 3. (3,000gp)
Standard Rod of Giant Summoning up to 6. (11,000gp)
Greater Rod of Giant Summoning up to 9. (24,500gp)
Have you read the text of the item or have you just heard about it from someone?
Yes you can, the construction requirements of a spell have no real interaction on the final product other than as thematic pointers.
The only reason you couldn't use the rod and feat together would be if they gave the same kind of bonus and didn't stack. But the giant simple template gives primarily size bonuses while augment summoning gives enhancement bonuses, no overlap.
First, yes. As a specialist wizard you can choose two different schools as your opposed schools. If you know you want to use a lot of polymorph spells then just don't select transmutation as one of those opposed schools.
Secondly, I think you might be misreading either the Merge with Familiar spell or maybe Familiar Melding.
Merge with Familiar lets the spellcaster merge the familiar into their own body. It doesn't let them merge the familiar into anyone or anything else.
Familiar melding is likewise locked down between the familiar and the caster, but uses the familiars body as the host for the caster's mind.
Of course I could be completely off and you could be referring to another ability I'm unaware of, but these were the closest two I could find from your description.
Umm, I'm not seeing that ability in the conjuration school powers or in any of the sub-schools. Could you give me a source or a link to where you found it?
OK, I found it. If you are referencing something from one of the non-core books it's usually good to either quote or cite the book in question so people know where to go looking.
Anyway, as the ability says, mental stats and special familiar abilities stay, everything else is overridden by the summoned creature.
So for the duration of the spell the combined familiar/summoned creature uses the summoned creature's HP. After the spell ends, by duration or banishment, the familiar is left behind with the same number of HP he had before the merge.
No, they're pretty much all in the animal archive or the familiar folio, both from the player companion line. Core are the main, big hardcover books that pretty much everyone knows relatively well, the smaller paperback lines tend to be less well known so it's nice to point out where you're getting the rules from.
So like, for your previous question, instead of just asking about what happens when your familiar melds with a summon, ask something like "When I use the Summoned Shell ability of the conjuration school familiar..." Or "What happens when I use the Summoned Shell from the familiar folio?"
Really, just using the actual name of the ability really helps since with sites like d20pfsrd and Archives of Nethys it's easy to look up abilities, but it's hard to search for something if you don't know it's name or what it is.
For instance, I tried looking up the words "meld" and "familiar" on both those sites and only found those two spells, since the Summoned Shell ability doesn't use the word "meld" at all.
Reference:
Greater Summoned Shell
/cevah