What general advice would you give to a newbie 1e DM?
With just the PHB and DMG, there are tons of things for which there are no rules, either in or out of combat. The UA was primarily character classes, spells, and magic items.
Players have no real skills based on these three books. The DMG has a list of craft skills, but that is downtime related for the most part.
There is no form of spot/search, unless you count a chance of detecting certain dungeon features, so you get to describe everything, and anything you don't, the players don't know about.
There are almost no combat maneuvers. If you want some, you have to make it up or say no when a player asks to charge, or trip, or anything like that. Technically speaking, it's just move, attack roll, attack roll, spell, attack roll, move, attack roll; and just hope that none of your players has psionics.
What particular rules should I look out for?
Good luck with figuring out the surprise rules.
There is no real system for handling for illusions. You'll have to house rule on the spot for many cases. 2e did a better job of it, you may want to grab those rules for 1e.
The random magic item tables can generate an artifact. I recommend ignoring that result. Some other things, like results of Staff of the Magi, or Vorpal Sword, should also be ignored for quite a while.
The Stone of Good Luck looks innocuous, but it's affect on the random magic items table, +/- 1-10 points on the roll as most beneficial, is good for the players and bad for the GM, as it tends to grab the most powerful and best items, especially artifacts, while avoiding all cursed items; which I never liked anyway, but they were a part of 1e.
In fact, it's best not to use the random magic item tables at all unless you're absolutely stumped, and probably not even then.
What adventures would you recommend?
For right after B1 - Keep on the Borderlands:
U1 - Sinister Secret of Saltwater (levels 1-3)
U2 - Danger at Dunwater (1-4)
U3 - The Final Enemy (3-5)
While maneuvering directly from wherever the distant borderlands are to the coastal community might be iffy, I still like these.
Would altering the race/class/level limitations game-breaking?
In the overall game milieu, the multi-class rules in 1e mean that races that can multi-class who do not have level limits will eventually completely eclipse humans (who cannot multi-class; and dual-classing is not that great). This is basically a matter of world design logic.
However, in an actual campaign, if you hand out even XP to everyone, it shouldn't be much of an issue in your actual game.
If you're in any milieu like Greyhawk or even Forgotten Realms, demihuman nations and their relationships and power levels versus human nations won't make much sense, but a lot of groups don't care about that type of issue.
What's the general opinion of the various "Survival Guides" or Unearthed Arcana?
The Cavalier and Barbarian classes in UA are overpowered, even though they are nerfed from their original Dragon Magazine presentations (when they were beyond uber). I recommend nixing them both.
Watch out for, and avoid, the Wand of Force in the UA. Unless you want lightsabers in your game.
The Dungeoneer and Wilderness Survival Guides had lists of non-combat proficiencies.
While a kludge in comparison to an actual skill and task resolution system, the non-combat proficiencies were better than nothing.
They also had lots of mechanics regarding climbing, moving through various terrains, food and water (and locating them), sleep, shelter, beasts of burden, weather, and survival in general. The DSG had an excellent mini-underdark setting in the back, including an equally excellent three-tier map. The only drawback to me was that getting around between some of the encounter areas can be very difficult, and the various groups were pretty isolated (some more than others). If you care about these types of rules, the DSG would be a good extension, and possible additional inspiration for expanding on, D1-3.
From the non-combat proficiencies in the WSG came Alertness, the only perception-increasing ability players can obtain that I can recall. It gave a chance to notice "disturbances", which altered the surprise chance. Now only if the surprise rules made any sense. It didn't matter though, almost every player took this one for his or her character (along with Blind-Fighting, which was useful as it reduced fighting penalities).
Although only a pet peeve of mine, for some reason, the WSG seems to think it takes one shepherd to handle six sheep on the move. That is utter nonsense. One shepherd can manage hundreds of sheep, and more than a thousand with a horse and a sheepdog.
I personally like both books, but remembering all those rules to start out with is going to be quite a task on top of learning 1e in the first place.