Did they actually look like that?

[Innocentius]

I 've noticed that the horse wears mail under the tunic. Does that mean that this was normal? I mean, I see pictures of knights with tuniced horses but I always thought that fabric was not a particularly good armor, but mail underneath would make a lot of difference.

Just because it is shown in this one single picture does not mean that it was standard equipment. Horses seldom wore any armour since this was very expencive, IIRC, arrmoured horses were rare even in the late medieval period. It is possible that more horses than we today know of actually wore quilted armour, and it's just that this doesn't show in the pictures from this time. But overall, I don't think horses had armour very commonly, as this would've exhausted the horses a lot faster.

[Watchman]

Armour levels changed seriously during the 14th century. Until around the middle of it most horses had quilt at best and maybe in rare cases a chamfron (head protection; "horse helmet" really) of something a bit more solid, and knights wore mail although increasing amounts of hardened leather reiforcements were being added on top of that. Assorted famous massacres where knights got chopped up, skewered or shot at way too much by commoner grunts plus the Hundred Years' War and a few technological refinements led to the mail being topped by first coats-of-plates and later individual pieces of solid plate plus substantial greaves and vambraces by the last quarter, and articulated full plate had made its entry by the last decades of the century. Horses, being large and vulnerable (at least in comparision to the increasingly untouchable rider) targets and prone to panicking when wounded, were similarly getting a lot more defensive gear.

By the end of the Hundred Years' War a man-at-arms with up-to-date gear would be protected by articulated full plate of tempered steel and mounted on a plate-barded horse. That's technick'ly what we calls a "tough nut".

Note that the level of defensive gear worn by lesser troops was on a similar upward spiral, although obviously always a few notches below what the top dogs could afford. This was partly because the rapid evolution resulted in knights constantly discarding older equipement to the market, and partly due to the aforementioned technological refinements which pushed prices down, but in any case great many infantry grunts were by early 1400s as well equipped as a knight of the late 1300s.

The divider between "light" and "heavy" cavalry was mainly that the former left off the large pauldrons of the full plate to improve the mobility of their arms and rode un- or lightly-barded horses, for example. A "light" infantryman differed from his "heavy" colleague chiefly in some details of weaponry, and wearing considerably less leg armour to improve mobility...

You wouldn't want to fight in a cloak, it could be used to pull you off the horse...equally, I don't whether spurs would be a wise idea when your horse is wearing curtains like that!

You'd actually be surprised how inconvenient stuff people cheerfully wore into combat without thinking twice about it. Merely as one example the heavy cuirassieurs of the Early Modern period almost invariably wore a brightly coloured sash across their body, whether as some sort of identificator or out of sheer soldierly vanity I don't know.
Period drill manuals bluntly recommend grabbing the enemy's sash and pulling him off the saddle by it if the opportunity presents itself. Which didn't keep anyone from wearing them.

As for the spurs, aside for being a symbol of knightly status (not that every horseman didn't use them) they were a very important tool at controlling the mount, especially when your hands were preoccipied with fighting. The "long stirrup" method of riding European heavy cavalry used (it has something to do with the use of couched lance, but I don't know the details) actually required developing a really long and inconvenient-looking type of spur for no other reason than to reach the flanks of the beast. Anyway, obviously neither armour nor caprison (the "curtains") covered the parts the spurs were supposed to press into. Although it might seem somewhat counterintuitive this didn't actually create a "weak point" in the animal's defensive gear - as the rider's armoured legs went right over that part.

[KrooK]

I have to agree with Watchman
Knights into that era were wearing coloured clothes on their armours.
It has some advantages;
- lower chance that someone from your army mix you with enemy (remember that enemy have similar armour)
- show opponent that you are rich and you can pay ranson, so they earn nice money if they don't kill you
- secure knight and his horse from sun (remember that wearing steel armour into sunny day might be hard experience - look at Grunwald battle) - especially into Holy Land or Spain

[The Stranger]

No lances were wood. Even the ones that had the fluted bit were made from wood. Carved on a lathe. The only metal bits would be the tip.

i know that... i was being sarcastic ;)