Non EU wife staying more than 90 days in EU

EU freedom of movement is codified in Directive 2004/38/EC. This has the answers to most of your questions:

How do I use my and my wife´s right to freedom of movement?

You can use your right to freedom of movement by going to any EU country other than the United Kingdom. (In the UK, you have a right of abode under UK law.) Your wife derives a right of freedom of movement from your EU citizenship whenever she travels with you or to join you. (That does not apply to the UK, where she would not be covered by the EEA immigration regulations.) The relevant part of the directive is Article 3, section 1:

Article 3

Beneficiaries

1.  This Directive shall apply to all Union citizens who move to or reside in a Member State other than that of which they are a national, and to their family members as defined in point 2 of Article 2 who accompany or join them.

While you do not seem to be "moving to" another member state, you are temporarily "residing" in several member states as far as Article 6 is concerned.

Do I point out at French Immigration that my passenger is my wife and as such her passport should not be stamped?

You may point out that she is your wife. In fact, you should probably carry a copy of your marriage certificate in case anyone asks you to prove that. For entry into France, that is unlikely, but after she has exceeded her 90 days in the Schengen area, the chance that you'd be asked for documentary evidence is probably much higher.

You should expect, however, that her passport will be stamped. The directive explicitly forbids stamping only for family members who have a residence card issued as evidence of their rights under the directive. Article 5, section (3):

3. The host Member State shall not place an entry or exit stamp in the passport of family members who are not nationals of a Member State provided that they present the residence card provided for in Article 10.

(emphasis added)

I assume that your wife has no such residence card because I infer from your question that she does not live in the EU. Therefore, her passport may be stamped.

Does [a Schengen country] have to let us in even though we have gone over 90 days?

Yes. The right of entry under the directive is established in Article 5. Here are the first two sections:

Right of entry

1.  Without prejudice to the provisions on travel documents applicable to national border controls, Member States shall grant Union citizens leave to enter their territory with a valid identity card or passport and shall grant family members who are not nationals of a Member State leave to enter their territory with a valid passport.

No entry visa or equivalent formality may be imposed on Union citizens.

2.  Family members who are not nationals of a Member State shall only be required to have an entry visa in accordance with Regulation (EC) No 539/2001 or, where appropriate, with national law. For the purposes of this Directive, possession of the valid residence card referred to in Article 10 shall exempt such family members from the visa requirement.

(emphasis added)

Because your wife is Mexican, she does not require a visa under Regulation (EC) No 539/2001, so all she needs is her passport.

But do the 90 days apply to her? The 90 day requirement is established in article 6 of the Schengen Borders Code ("entry conditions for third-country nationals"). In Article 2(5)(a), it includes family of EU citizens in the term "persons enjoying the right of free movement under Union law". Section 6 then excludes those people from the term "third-country national":

5. ‘persons enjoying the right of free movement under Union law’ means:

(a) Union citizens within the meaning of Article 20(1) TFEU, and third-country nationals who are members of the family of a Union citizen exercising his or her right to free movement to whom Directive 2004/38/EC of the European Parliament and of the Council ( 1 ) applies;

6. ‘third-country national’ means any person who is not a Union citizen within the meaning of Article 20(1) TFEU and who is not covered by point 5 of this Article;

(emphasis added)

This means that Article 6 does not apply to your wife when she travels with you, so there is no 90-day limit.

Furthermore, her presence in Schengen countries is controlled by the freedom of movement directive, which has this to say about the "right of residence for up to three months." Here is the entirety of Article 6:

Article 6

Right of residence for up to three months

1.  Union citizens shall have the right of residence on the territory of another Member State for a period of up to three months without any conditions or any formalities other than the requirement to hold a valid identity card or passport.

2.  The provisions of paragraph 1 shall also apply to family members in possession of a valid passport who are not nationals of a Member State, accompanying or joining the Union citizen.

This means that you and she can stay together for up to three months in each EU country. If you want to stay in any one country for more than three months, you can refer to Articles 7 through 15, which discuss the conditions under which you can do that, along with the (mostly relatively mild) penalties that can be imposed for remaining for more than three months while failing to meet those conditions. Your description of your plans suggests that this will not be likely for you.

You are probably concerned about how your border crossings will proceed. This is hard to predict, because different countries may have different procedures even within the framework of the common Schengen Borders Code. I can offer the experience of my parents, one of whom is an EU citizen and the other is not. They have spent 4 months in the Schengen area on a number of occasions, generally entering and leaving through France. The French border officers did on one occasion become somewhat concerned by my mother's apparent overstay in Schengen, but as soon as they realized she was married to my father, they sent her on her way. After that time, they started approaching the passport control desk together, and they never had a problem after that.

With that in mind, you may be interested to know that you and your wife should be accepted together at the EU passports desk even though she does not have an EU passport. Remember that Schengen Borders Code includes her as a "person enjoying the right of free movement under Union law," and then turn to Article 10, section 2:

2.  Persons enjoying the right of free movement under Union law are entitled to use the lanes indicated by the sign shown in Part A (‘EU, EEA, CH’) of Annex III. They may also use the lanes indicated by the sign shown in Part B1 (‘visa not required’) and Part B2 (‘all passports’) of Annex III.