It is my understanding that for a Protestant baptism to be valid, there must be an intent to do what the Church does. I would assume this would mean that there is an intent to instill sanctifying grace and remove original and any later sins. So this would make most High Anglican and Lutheran baptisms valid but illicit. Prior to Vatican II these converts would still be conditionally baptized. If the intent is only to do what they think the "church" does, and the Protestants think are the church, this would mean that so called "believer's baptisms" are valid which is certainly not the case. 

It is my understanding that for a Protestant baptism to be valid, there must be an intent to do what the Church does. I would assume this would mean that there is an intent to instill sanctifying grace and remove original and any later sins.

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No, not even.  One needn't believe such things (grace, sin, etc.) even exist, ergo the validity of baptisms performed, say, by an atheist midwife or a Jєωιѕн doctor.  Intention has to do with intending to do the actual act, not with what one thinks the act does (or doesn't) accomplish (this is the same reason that, say, a priest who doesn't believe in the Real Presence can still effect the Eucharist).  
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The standard for intention of 'to do what the Church does' is admittedly vague, and that's because the bar for sacramental intention is itself incredibly low.  The intention of an emergency worker (who isn't Catholic, and might not even be Christian) who issues an emergency baptism at, say, the request of an incapacitated mother whose newborn is dying, is probably just intending something like 'I intend to do this thing Christian people do to their babies' and that suffices.
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Now, when it comes to conditional baptisms after such baptisms (or after even baptisms performed under more 'ordinary' circuмstances, like the baptism a child might receive from Lutherans before later converting to Catholicism as an adult), such conditional baptisms have more to do with simply lacking any evidence that the baptism was performed, or performed correctly.  In lieu of positive evidence, the Church conditionally baptizes.  
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Technology makes it remarkably easier to track down the sufficient evidence.  Baptismal records are digitized which makes their archival and retrieval far less susceptible to human error (this of course assumes the baptism was recorded in the first place).  If the minister is still alive, they can be interviewed as well, just with a quick email or phone call.  Photographic evidence is extremely common, and video evidence is becoming even more common.  But, try to track down a Lutheran minister who performed a baptism thirty years ago in, say, 1899 and you'd have a much more difficult time.  Hence, the preponderance of conditional baptisms.
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I hasten to add that even with all of this said, I don't think that it would be correct to say that there is a presumption of the validity of non-Catholic baptisms (maybe there is, I've just never seen anything to indicate that's the case per se).  Such baptisms should still be investigated.  If the investigation does not satisfy all reasonable and positive doubts, conditional baptism is performed.

No, not even.  One needn't believe such things (grace, sin, etc.) even exist, ergo the validity of baptisms performed, say, by an atheist midwife or a Jєωιѕн doctor.  Intention has to do with intending to do the actual act, not with what one thinks the act does (or doesn't) accomplish (this is the same reason that, say, a priest who doesn't believe in the Real Presence can still effect the Eucharist).  

In the True Rites of the Church, She made the form of the prayers of the sacraments CONTAIN the intent of the Church, so that, as long as the correct form/prayer is said, then the intention is correct.  Thus, if the prayer is properly said ("I baptize thee in the name...", or "This is My body") then the sacrament is valid.  For the True Mass, even if a priest doubted the true presence, or even became an atheist, the consecration would be valid if he followed the prayers exactly.
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This also applies to the new rite of baptism because the prayer was not changed.  So if the prayer is said, then the baptism is valid.
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However the form/prayers of the NEW MASS have been changed, as Cardinal Ottaviani (one of the top theologians in rome in the 60s) has pointed out.  So, in regards to the new mass, the priest must supply the proper intention for the consecration, or it is invalid (or it is doubtful that it is valid).  So, for the new mass, if a priest doubts the real presence, then the consecration is probably invalid.

 

In the case of an athiest midwife, for instance, would not she be intending to do what the Catholic Church does for that baby, despite her unbelief? 

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Yes, which is all that she needs to intend (of course remember that matter and form must be present as well, so she still needs to say the correct words and use water). 
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Remember that the axiom for intention is to intend to do what the Church does, NOT to intend what the Church intends.  A subtle but absolutely crucial difference.  This is the same reason why there is no prima facie reason to doubt, say, holy orders that have a freemason somewhere on their lineage.

By your logic Evangelical baptisms would be valid correct?

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I don't want to claim that baptisms of any non-Catholic group are categorically valid, since the Church has never said that.  Baptisms issued outside the Church should still be investigated.  Only, the incorrect or heretical beliefs about baptism on the part of an Evangelical minister simply have no bearing on whether or not a baptism he performed is valid. 

For the priest to validly consecrate the host, must he believe in the real presence?

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No.  Faith has nothing at all to do with the validity of effecting sacraments goes.  This was settled almost two thousand years ago with the Donatist controversy.

Isn't the act itself the intention?  Prior to Vat II all nurses in the maternity ward were instructed on the act of baptism in case of emergency regardless of their beliefs.
The same for ordinations.  If the Bishop publicly performs the act of ordination regardless of his personal state of grace or even evil intentions the candidate is still ordained.

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When it comes to Catholic ministers, the rule is that so long as the act is performed with the right matter and form, the intention is presumed and therefore the sacrament is presumed to be valid. 
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I do not think the same rule applies, per se, to non-Catholic baptisms.  When it comes to midwives who baptize, they were trained to baptize because of Catholics, so far as I am aware.  Remember that Catholics are the only Christians who insist on the necessity of infant baptism; even Lutherans who baptize at a young age, i.e., before the age of reason, still don't actually believe it's necessary for salvation.  So, it isn't as though the Lutherans would have been exerting pressure on medical professionals to baptize.  The point is that the knowledge of midwives and doctors who were trained to baptize, that training was ultimately from the Catholic Church.  And that is why any presumption of validity could be had.  At least, that is my understanding-- I've never seen it put explicitly that way, it's just what I've come to reason is the case reading about the issue in various contexts.
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Now, when it comes to baptisms performed under ordinary circuмstances by non-Catholics, their rites of baptism are not supervised by the Church.  The Church does not instruct those ministers on the form, the matter, or anything else.  That is why the Church investigates those baptisms when she receives converts from heretical sects.  And if the investigator finds out that the baptism checks out, no conditional (or absolute) baptism is necessary.  But if no sufficient evidence can be had (or, if sufficient evidence can be had that actually demonstrates certain invalidity), a conditional baptism is done.
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Actually, they do baptize in the name of the Father, and of the Son, and of the Holy Ghost --- but each of those terms mean different things to them, than they do to Catholics or other Christians.  Their whole belief system is based upon appallingly false concepts of God, and they can only be called Christians if you define that term as "someone who acknowledges the spiritual significance of the man called Jesus".

Mormons are not the best example to use for this discussion. In pre-V2 thinking, a baptism using water and a proper formula should be considered valid, and depending on investigation, result in either a conditional baptism or supplying of ceremonies for converts. It has been said above even non-believers can validly baptize in an emergency. But the post-V2 church considers Mormon baptisms using a Trinitarian formula invalid because Mormons are non-Trinitarian and mean different things by the words of the formula.

Our local hospital was owned and operated by the Sisters of St Joseph where they trained nurses.  I'd imagine there was a nurse/nun in the delivery room and it was actually she who baptised infants in danger of death.

Even a Jєωιѕн rabbi, if he intends what the Church intends, and does what the Church does can validly baptize someone.

But why would a Jєω rabbi Baptize, when his тαℓмυd tells him the Gentiles are cattle and Our Lord is in Hell?