"The dragon of Revelation 12 and the talking idol of Revelation 13."
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Question #1:  
Regarding  Rev. 12:17, “The dragon grew angry with the woman, and went away to make war with the rest of her seed, who keep God's commandments and hold Jesus' testimony”.
Is the dragon, Satan?
Is the woman, Israel?
Is the seed, Israel or all believers in Christ?
Response #1:  
This is how I take this passage with some clarification on question three. 
The dragon is Satan - he clearly carries out this persecution through the agency of antichrist, the beast.  Note that both the dragon and the beast are 1) red, 2) have seven heads, 3) have ten horns.  These events take place at the Tribulation's mid-point.  When the two witnesses are killed (see Coming Tribulation part 3A), those in Israel who believe become the objects of antichrist's persecutions (preliminary efforts: Dan.11:28; this persecution: Dan.11:32-35; also Dan.8:23-25).  This marks the beginning of the "Great Persecution" which will start with Moses and Elijah and the 144,000, spread to the rest of believing Israel who are in the land (the passage you are asking about, Rev.12), and then, after their escape into a divinely prepared desert sanctuary, shift quickly into a world-wide persecution which will be directed not only at believers of Jewish heritage but at all who "obey God's commandments and hold fast to the testimony of Jesus".  The waters spewed out of the dragon's mouth are some of the beast's armed forces sent to destroy the Jewish believers present in Israel at the conclusion of the second campaign against the southern alliance (covered in Dan.11:29-30); it is also at this time that antichrist abolishes the temple rite and sets up the talking idol in the temple court (the "abomination of desolation"). 
More details for the entire context are contained in the very soon to be released part 4 of "Coming Tribulation".  Hope this helps fill in some gaps in the meantime. 
Yours in our Lord and Savior Jesus Christ, 
Bob L.
Question #2:  
What is the 'talking idol'?  Is the talking idol a person or what?
Response #2:  
The "talking idol" is referred to in Revelation 13:14-15, and called "the abomination of desolation" in Dan.11:31; 12:11; Matt.24:15; Mk.13:14 (better translated: “the abomination that causes [spiritual] desolation”).  This is a life-like statue of antichrist placed in the temple court, that is, the inner court (cf. Rev.11:2 - Greek, wrongly translated in most versions).   Here the unbelieving world will pay its respects to this image of antichrist that appears to be alive (but is not).  Antichrist himself will hold court within the temple, taking his seat in the holy of holies "as if he were God" (Dan.8:11; 2Thes.2:4; Rev.13:11-18).   The time frame for all this is the Great Tribulation, beginning at the Tribulation's mid-point and lasting for "2,300 evenings and mornings" (Dan.8:14) until antichrist departs the land of Israel prior to the Armageddon campaign (as a result of insurrection occasioned by the "darkening of his kingdom"; Rev.16:10; cf. Dan.11:44-45; Zech.12:6-9). 
You can find out more about the statue and related issues at the following link:
Part VIII of Coming Tribulation 3B:  “The “Abomination of Desolation” and the “Session” of Antichrist” 
In Him who is the truth, our Lord and Savior Jesus Christ. 
Bob L.
http://ichthys.com