Lots of questions from a newbie LotFP referee

1) About the "Paralyse" saving throw.

There's a balance between classes in saving throws. For example, the halfling has the best saving throws and that's their main class feature. They are mechanically balanced, and anyways, there's no explanation of how they work in fiction, so you can make it up. Maybe the magic user has better saving throw when avoiding to fall into a pit because he has an uncanny sixth sense. Before tinkering with a game, always try it as it is first.

2) A person with Constitution 10 travels 24 miles a day, and a person with Constitution 18 travels 27 miles a day. Is this correct?

I'm prone to think it is, yes. Not many words are spent on the subject but since you add your constitution modifier, it should be a correct assumption. Notice that if your group travels together, its speed is going to be that of the slowest moving element of the group. Or do you leave people behind when you travel?

3) The movement table (Rules & Magic, pg. 38) lists exploration, combat and running distances per round. What is the normal, "strolling down the street" distance per round in feet?

I'd say the average movement as listed in the table.

4) About the skills. I've been thinking about using difficulty modifiers, such as "+1 for an easy climb". Is this something that other referees do? My motivation is to prevent silly situations like a Fighter repeatedly failing to climb a 9 feet tall wall in a situation in which I cannot rule that no dice roll is required (for instance, running from the city guard and attempting to climb over a wall in an alley). However, I'm worried that this might make the specialist class' skill focus seem less important overall.

You're the referee and you are ALWAYS entitled into ruling that no dice roll is required. In fact, it should only be required when you cannot decide and a random result is needed. So making a character roll for an easy climb makes no sense to me. Or if you roll and fail, that might only mean that you take longer, not that you fall. It's an old school game. Ruling, not rules! The book states it clearly that you, the referee, ARE the system. You decide when dice are rolled.
And you're right, it'd weaken the Specialist.

5) Concerning XP rewards for the sample adventure [SPOILER alert!]. I very much expect my players to never even find out about the jewels unless I'm very heavy handed - "hey, there is TREASURE inside some of these people! Hack away!". Is it "ok" if my players end their first adventure with 0xp? Do you think it will be too frustrating?

I think it would, yes. There should always some loot, or they'll never advance. It's not difficult to include a treasure. If you think your players are never going to find out about the jewels, then maybe you should place them in an easier position for them to notice/reach.

7) Concerning some developments in the sample adventure [SPOILER alert!]:

I'm afraid I cannot reply to this question because I haven't played the adventure yet, but I may get back to it in the future if I have an occasion.

8) Holy Water (Rules & Magic, pg. 60) seems like a lot of effort for almost no gain (10 bless spells, plus a silver vial, for a 1d8 damage missile). Am I looking at it the wrong way?

No it's correct, it shouldn't be too easy to obtain one, or was your group's cleric hoping to walk around with a hundred vials strapped to his belts? Besides, the uses are limited, not all monster should be vulnerable to blessed water, so it's not something to worry about too much. And besides, think of all the narrative consequences of having blessed water. It could be used to heal illness, or do something equally cool. Your choice as Referee, but it doesn't have to be just a weapon.

9) Critical hits. Does anyone use them?

No, I use ruling. I never assume that a character just hits or misses the target, and always add something in fiction that makes every strike interesting. I expect as much from the players, so there's no real need for critical hits. The '1 is always a miss and 20 is always a hit' rule already works well enough, but if you want to add extra spice, funkaoshi gave some valid alternatives.