As expected, it's really about not letting people have pleasure

[Black Aeronaut]

Sorry BA, it is neither reasonable, nor rational; the effort is there, but it is deeply, fatally flawed on multiple levels.
Process of Democracy - specifically in law is that the law is applied equally to everyone.  So your proposal can be interpreted two ways.

• The first is that the blastocyst.fetus is a person.  This fails in that, based on you plan, not all all treated equally.  Some are protected (following the third strike) while others are not.  This breaks Process of Democracy.  It also does not answer any questions as to the rights of any fetus being carried to term by an individual.  I will address this point further later as it is very important.
• The second is that you are punishing women for their reproductive choices.  This is just misogynistic;  I know you mention the role of men later, a point I will be coming back to. 
  
• Neither of these points addresses 'morning after pills' and similar after fertilization contraception.  Are these mechanisms Class A strikes, Class B strikes, foul balls or another baseball analogy?
  

Okay, so will get to Point 1 later.  Point 2... Whoah.  So it doesn't strike you as irresponsible at all that a woman that is fully capable of taking care of a child decides to get an abortion instead of carrying the child to term and putting it up for adoption?  You see, that's what really sickens me - that someone will get an abortion instead of putting the child up for adoption.  And it's not like a woman would lose her job over a pregnancy - unless I'm mistaken don't most states have laws about that sort of thing?
As for Point Three... further on down.

Strikes WILL count against teens that have consensual sex. Even if they
used contraceptives - if they're using them then they should know the
risks.

So you are treating teens like kids in regards
to voting, drinking, and crime, but take special exception for their
reproductive behavior?  That is just creepy.

No.  What's creepy is what you say next...

You are also making the
rather egregious assumption that they are informed and know the risks. 
There is a woefully large percentage of the American youth population
that has been let down by their educators in regards to sex ed.

That parents and educators both fail to teach their kids that there are serious consequences for having sex?  THAT is creepy.  And I'm not even gonna touch that whole contraceptive implant thing.
However, upon further reflection, I can see your point.  These things will have to be investigated carefully, and in the cases where one party honestly didn't know the consequences, they'll get a pass.

Also, the male parent gets a Strike as well, whether or not he's in support of the abortion.

Which requires forcing a paternity test on all abortions.

Not like there hasn't ever been legal precedent for paternity tests before.  Although we can say that another man may claim responsibility if he chooses to.

Why so stringent? Because if he's sleeping with a woman then he knows
the risks as well. He needs to be held responsible for that - even if
the woman gets the abortion against his wishes (she could just give the
child over to him to care for and leave him if she doesn't want to take
care of the child).

This breaks down in practice.  You are
holding an individual responsible for things that they have no control
over.  If you are punishing the male parent, they should have choice in
the matter; being able to force the pregnancy to term - which is again,
creepy in the extreme.  I will assume that your class B strikes also
apply to the father?

Look, a man does have a degree of absolute control over this issue.  That control is, "Do I or do I not stick my dick up inside this woman?"  (For crying out loud, even super-star rappers sing this... "Eighteen years!  Eighteen years!  She got one o' yo kids fo' eighteen years!"  Different point there, but still: consequences.)  Sure, the two may be looking to just have a fun little fling, and sure, accidents do happen... but when an accident happens you have to accept the consequences.  It's no different from any other right/privilege.  Drive your car and hit a pedestrian?  It's an accident, but you're stuck with the consequences.

Practical considerations - National Abortion Database?  Monitored at the state or federal level?

*Blinks*  Good question.  I'd say state level so each state can put their own spin on it.

To further clear up moral ambiguity, Strikes can be divided into two
classes. A 'Class A' Strike is the nastier sort. This is given to
mothers who are healthy, are pregnant through consensual sex, the fetus
is biologically viable, and they have the means to care for the child.
In other words, they have no excuse save that it would be
'inconvenient'. Three of these and you face prosecution.

So
how do you judge means?  Again this goes against Process of Democracy. 
Income?  Number of extant children?  Marital Status?  Psychological
Profile?  Employment?  Education?  Any or all of the above?  Whatever
mechanism you use, you will have to be able to apply it consistently for
every case.  I should also note that every single abortion would have
to be subject to judicial review; as your three strikes law would
necessarily have to have some sort of sanction for those reaching the
third strike - chemical of physical sterilization, imprisonment, etc. 
Oh and all miscarriages and stillbirths will also have to subject to
examination and review to ensure that they were not deliberate attempts
by one or more parties to be artificially induced.

... At least we can agree that abortion is not something to be taken lightly.
As I mentioned before, this involves government getting involved to an uncomfortable level.  Yes, this means that the government is going to put you under a microscope.  But this is no different than when you're applying for state welfare and they ask to see paystubs and bank statements.  Also, this would be why I feel it would be better if applied at the state level, because the living standards/cost of living in Texas are hella different from other places like... say, California?  Also, each state can work out its own punitive measure.  I won't take any stance on what that measure is.

We can also eliminate the need for the state getting involved like a welfare service does by going the other way: force a woman to carry to term, and then put the child up for adoption through the state system.  This would also eliminate the 'Class B' strikes altogether.  Which method squicks you less, Rev?

Your
proposal also does not really address whether or not the
blastocyst/fetus/baby is a person under the law; or at what point they
become a person under the law.  From your description it is black or
white.  Abortion is abortion if carried out three days after using plan B
or at nine months.  Personhood under the law is hugely important in
this debate is hugely important because of the implications to the
mother.  For example, if a fetus is a person under law, then any
behavior by the mother, such as drinking, recreational drug use, or even
medical treatment for existing or developing condition - anything that
put the fetus at risk could be subject to legal sanctions.  Children are
removed from abusive households - how would such things be handled
similarly for equivalent actions during pregnancy.  Is the woman taken
into custody for the duration?

That's because I wasn't really intending to deal with that aspect of the law.  Let someone else define what the legal definition of abortion is.  This is just to deal with that definition once it gets sorted out.  (To be honest, I don't know if there actually is a standing legal definition of what abortion is - until you brought that point up I thought there was.)  But, for the record, let me say that I would tacitly lump in day-after pills with contraceptives.

Abortion is a difficult topic; there is no right answer, or best answer, all we can hope for is the least worst answer.

Indeed.  No matter how you deal with abortion law, it's gonna be extreme in one way or the other.

If everyone has the right to abort when they damn well please?  You're killing babies!  Evil! 
Make abortion illegal?  You're nixing people's rights! Evil!  Get the
government/people involved in other peoples' affairs?  Evil! ... but probably a lot less so than the other two options.