I love hearing other people's reasoning about their headcannons, etc. 
Yes, Meredith seems inconsistent when it comes to Anders or a mage Hawke. But Alrik, I've kind of assumed was part of what went on that Cullen tells you in DAI that Meredith kept from him. She may have turned down the proposal but turned a blind eye to him turning some mages tranquil because she thought the fear of him would keep other mages in-line and she could rein him in at any time. Or he simply accused them of blood magic to her and she took his word repeatedly?
Yeah, this is turning out to be a really fun conversation. Thank you for that. Right now, with what I'm writing I've got Meredith's stance on Anders being that he's a known Warden and she doesn't want to public tangle with the Order just after a Blight. And the fact I can't see the Wardens discussing internal conflicts with anyone, I doubt they'd give Meredith the time of day.
I also am working with the idea that she's not ready to march through Darktown, killing all the people who'd stand up for their healer. It'd be hell on the Order's reputation. That doesn't mean covert action can't be taken, but screwing up with Karl and letting him know that they really were onto him was a bad idea. I would have just sent disguised Templars to the clinic after hours myself, on some excuse and then take him.
On Alrik, I can believe he'd claim blood magic, and that put Meredith into the willfully blind category, imo. Again in said story (and have I mentioned writing Meredith is a pain in the ass? We get 10 minutes of not so crazy, then Act three) her official reasoning for not supporting it is that the Circles can't get every mage, making them Tranquil immediately would be worse. Parents who might be willing to hand their child over may be less inclined to do so if they're going to be made emotionless automatons. And if word really got out the Order was going to do this, the Circles would rise up because they'd be backed into a corner. Unlike what actually happened, the idea that every mage alive would be made Tranquil would give mages a unity of purpose that would be frightening. There'd be no dissent.
I know they officially say they're looking for unshakable faith in the Maker, but in practical terms, they mostly take young kids who are forced into the life. Obviously they get a few who wanted to be there - Lysette from Haven seems to be one. I think there's a huge amount of room for interpreting what is and isn't considered "moral" under the Chantry, though. I mean, this is an entire religion based on a god falling in love with a married woman (which has always left me questioning why they defaulted to a celibate priesthood when the origin of the religion has a lot more in common with some classical religions.) But while there are references to illegitimate children in DAO, it is only in reference to nobles' children because they aren't in the line of succession and with the exception of Alistair, the noble parents seem to be pretty open about supporting them. Sebastian got into trouble, but that seemed to be more because of public excesses that embarrassed the family rather than simply drinking and womanizing. 
Non-mage Trevelyan, iirc is mentioned that they'd either be going to the Chantry proper or the Templars, being that they're a fourth child and thus... shrug. I can also see, though this hasn't been mentioned, children of commoners being sent off if they've one too many mouths to feed. I'm not sure if that would truly work since Cullen's acceptance into training implied some sort of sponsorship, but I may be reading too much into that.
The Codex is interesting since it's not written by a member of the Chantry or the Order itself, it's written by an Enchanter, who opens the thing with the public perception of Templars by the common folks is that they're paragons of all that is right and true, but in reality, his implication, is that the Chantry prefers zealots because zealots are less likely to question.
The weird thing is celibacy was actually a late edition to the Church in the Middle Ages. They added it to the vows when Bishops and other high ranking members of the Church were willing their positions to their children.
The RCC is the only Christian denomination I remember adhering to celibacy as a rule. I think the Lutherans jettisoned that right away, though the Anglicans may not have. It's been a while since my comparative religion classes.
Of course the whole thing is that priests are married to the Church thus taking the place of the sacrament of marriage.
Agreed on the gossip - Carroll will make a remark about "stories" he heard in quarters from other Templars. But stoic - think the US Marines standing guard at the Tomb of the Unknown Soldier at Arlington or the Royal Guards at Buckingham Palace (do I have that right, lol?) That's because of disciplined training of how to behave on duty (anyone who's ever been through basic training and had to stand at attention and show no emotion while being screamed at can attest to that
.) The marines as a whole don't exactly have a reputation for only drinking tea when off duty. The more disciplined and controlled they are in public, the more they need some outlet off duty. And the other side of that discipline is unit bonding - that's where I'd imagine slaps on the shoulder while passing around drinks and stories in the evening and even an arm around the shoulder off duty after someone's first "blood" to help them adjust. I don't imagine a lot of physical contact, but something.
I have a lot of friends who are Army or Air Force . And the things they say about Marines... But yeah, they don't have a reputation for being straight laced off duty. I mean bar fights and gallivanting while on leave are stereotypes about military life in general. Plus I do live near Great Lakes Naval, so I run into guys who are either in or just finished basic going to the movies or Great America etc.
As for the soldiers at the Tomb of the Unknown Soldier. I have amazing amounts of respect for those guys. I've been to Arlington once, and the place gives me the chills. All that history, and doing what they do for hours on end. Wow.
I admit, I originally understood the Templars to be more like medieval knights, trained as individual warriors, not as soldiers, a very different thing. But they've made it clear that they're an army, otherwise Cullen would make absolutely no sense as Commander of the Inquisition because he wouldn't have been trained in all the complex things that are required for running an army...which he clearly does know quite thoroughly, up to siege tactics. You can even overhear a drill sergeant chewing out recruits at Griffon Wing Keep (which just happens to be commanded by Rylen, also Templar trained.) Those aren't things you pick up from a book in a few hours, nor is the mindset of an individualistic knight anything like that of a unit-minded army officer. But I also admit I'm evaluating it as ex-military with a strong interest in history and that doesn't mean the devs are writing it that way. For all I know, they're thinking "sword in hand, obviously they know the rest by osmosis." 
No, I agree with you. And while I'm not ex military, I've spent a lot of time with people who are over the years, and you can't help but pick things up. Plus one of my favorite reads is a hard military sci fi, which one isn't going to be able to follow unless you get the mindset.
I love that drill sergeant, he reminds me of the DI in Mass Effect 2 just outside the Customs station:
I'm not sure if the Devs really knew what direction the Templars were going to take until DA2. That's when I started to get the feel that they were more soldiers than warriors. Of course their rank system makes no sense what so ever. Unless you're Star Trek Original Series. They did't introduce enlisted and NCOs until O'Brien became a true reoccurring character on STNG. And even in the beginning he wore the rank pips of an ensign.
As far as I can tell, there's the basic knight, the lieutenants, captains and then commanders. After that it gets fuzzy again since they mention the Knight Vigilant and imply he's some type of high rank, but that's it. And then there's the Lord Seeker and the ability to take command of the Order even though the Seekers are separate and supposed to be something of an IAD for the Templars. That worked well, didn't it?
Obviously if they are a functional army they've more organization than we've seen, but it's not like the Devs can fully show us the extent unless they take the field, and even then, since we see the Chargers in Demands of the Qun and Bull's introductory quest, and I wouldn't call only Krem, Stitches, Dalish, etc the whole of the Company. However doing a hundred or so man battle for a small quest like this would be overkill and a waste of resources, all things considered.
I do wish we had more, but the only place I could see them giving us a detailed encyclopedic kind of thing would be in the RPG, and I'm not certain if that's even being published anymore. Plus I'm not certain how canon it is since I've seen mention in the game that Fenris isn't the only one with Lyrium tattoos. I'm not sure if they the put it in as a game mechanic, but it strikes me as a way of saying "sure you can play a PC with this!"