I. Are there any interesting differences between running campaigns for smaller vs. larger parties? If so, what are they? What constitutes a small or large group?
Party size is everything. It's so important, that I'm inclined to say that the entire 'Indy Gaming' movement was inspired running games for by small party sizes.
1) 1 Player - You can go into as much gritty slice of life detail as you like. The game is primarily about the characters emotional and moral change, and the PC can have very intimate meaningful relationships with NPCs. Player agenda is the absolutely most important part of the game.
2) 2-3 Players - You can have some gritty slice of life detail. The game is primarily about the relationship between the PCs, and a few NPCs that the group relates to. Play involving the relationship between one PC and an NPC is difficult, and really you are limited to only a few cases. Player agenda is important, but some concession must be made to an agreed upon purpose of the group. Sand box and dynastic play is very viable.
3) 4-6 Players - Gritty slice of life detail is less and less important and generally has to be handwaved to off stage events unless it is of the greatest importance. The game is primarily about the goals of the group, which may or may not be largely set by the referee. NPCs serve typical game roles like foils, mentors, quest givers, and enemies, but there isn't a lot of time to explore relationships between the party and the NPC - much less the individual. Intra-party dynamics is increasingly less important compared to pursuit of the overall group agenda, and individual player agenda's must take increasingly small portions of play. PC's may evolve in terms of their role within the setting, but evolution of the PC's emotional or moral life is largely a secondary consideration. Intra-party conflict is less and less viable as a focus of play, and much more likely to be anti-social and destructive.
4) 7-10 players - Group agenda is everything. There is almost no time for exploration of individual goals. Because such a large group likely can't agree on an agenda, pure sandbox play is increasingly unviable and group agenda is largely set by the referee's plot. NPCs however fleshed out and interesting largely have little role in shaping play other than as plot mechanics. By and large, only play that can easily be shared as a duty across the whole group - such as combat - is really viable. Long sessions of RP or other sorts of play draw too much spotlight to single PCs and with it, single players.
5) 11+ players - Chaos reigns unless the DM maintains the strictest discipline. A party leader has to be officially created to act as caller and mediate much of the groups collective interaction with the DM. The DM is basically forced into an 'us against them' format, and largely acts as the parties chief foil and adversary. It is necessarily to tightly govern the scope of play as simply getting anything done with such a large group is difficult. Since you are likely to have a rotating cast of players, some trope of play explaining why this PC or that is not available needs to be created or else there needs to be a very rigorously enforced social contract with players designating who has control of their PC when they aren't available. It is strongly suggested to adopt an old school style Haven/Mega-Dungeon structure. Close reading of the advice in the 1e DMG is strongly recommended, as it will suddenly start to make sense with groups this large.
II. Do you have an ideal number of players you like running games for? If so, what about that number makes it ideal?
4-6, but smaller groups allow for more intense role-play and more literary styles. Compare the aspirations of Tolkien in 'Fellowship' to his aspirations when the story is reduced down to Frodo, Sam, and Sméagol for an interesting analogy.
Bonus question: What are your overall feelings about running an NPC with your party? (good idea? bad idea?) If you've done it, what things did you like and not like about it?
NPC's shouldn't steal spotlight ever. An NPC with the party should largely be an appendage of one of the PCs of some sort.