General Spells and Their Levels

Does a General spell have to be taken at a "normal" spell level value? That is to say, between 1-5, or a multiple of 5? Can I have a level 21 Aegis of the Hearth? A level 8 Ward against Spirits of Air?

you certainly can.

In most cases it's not really useful, but you still mechanically can do it.

In previous editions, you certainly could take tham at absolutely any level you felt like.
In the current edition, you might actually, technically, be restricted, because of how spell design work.

that stated, my group has historically allowed it, and it didn't break anything, so go for it.

What this edition actually says is

things like wards often benefit from just one more level. In fact there is to my knowledge no statement that it is even appropriate to round levels to the nearest integer. I haven't rounded in my games, but I've yet to have a situation where fractional levels made a difference.

Very rarely could I see fractional level mattering. An extra fractional bit wouldn't matter for penetration unless things start having non-integer magic resistance. Most guidelines follow the integers and all Range/Duration/Target adjustments do, so it would rarely matter for those. But I can imagine a few instances. I suppose an SG might rule something that goes up by, for example, +3 each 5 levels actually goes up +3/5 per individual level, so if you want +1 you might add 5/3 of a level. I wouldn't do that myself, but it could be done. More typically, an integer plus 0.5 might make a difference when you double the level, such as with some Vim guidelines. Let's say you have a lab total of 43. You could make a level 21.5 spell in one season. Let's say it's Touch, Momentary, Individual and uses the first MuVi guideline in the core book. It would then work on level 43 spells instead of only level 42 spells for the level 21 version. But how many level 42 v. level 43 formulaic spells do you have that you want to cast this upon?

Still, as happy as I am with serious mathematics, I wouldn't want to deal with all this in the game. I'll just stick with integers.

My apologies. I was mis-remembering the content of "Changing Ranges, Durations, and Targets" p. 114.
In that case, the answer is simpler: You certainly can have Gen spells at odd levels, assuming the base level it 6+.

I'm skipping a step, It's simpler than integers.

Yes, at level 5 or lower the whole level/magnitude thing can yield odd results.

But in practice most spells should end with levels as a multiple of 5, and I only see General spells benefit from odd levels - because they are compared to Might or spell levels