Since you move the spell effect there is no reason why you cannot choose its path

There is no general rule that covers all spells whose AOE you can move. However, the two examples you specifically ask about can be addressed here sufficiently.

The description of the healing spirit spell (XGtE, p. 157) says:

you can move the spirit up to 30 feet to a space you can see.

This effect is very clear, you can move the effect 30 feet. It does not have to be in a straight line so, you can take any path you want.

The description of the moonbeam spell says:

you can [...] move the beam 60 feet in any direction

This one is perhaps a bit less clear, but a plain English reading would suggest that there is nothing restricting the movement path really. It doesn't say "any one direction" just "any direction".

This intent is confirmed in an unofficial tweet by rules designer Jeremy Crawford:

5e question: Moonbeam can be moved UP TO 60 ft. or EXACTLY 60 ft.? Other spells specify UP TO, Moonbeam doesn't

The intent is that you can move moonbeam's light up to 60 feet.

So again, you have complete control of the spell effect's path.

However, do note that this can be different depending on the wording of each spell. Some might restrict the path that can be moved even if these two don't. There is no general rule that covers all spells whose AOE you can move.

Moving an AOE that requires entering over a creature does not cause them to be affected by it

The Sage Advice Compendium has some explanation about AOE spells like moonbeam and how they affect enemies:

Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn't count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.

And, stated another way by Jeremy Crawford on Twitter:

Movement of Moonbeam not count as creatures entering when spirit guardians moving into creature counts as entering?

Moonbeam, spirit guardians & the like work the same way: a creature, not the spell effect, does the entering.

So, no, no matter how you move the spells you cannot affect more creatures.

It seems to be that the Orb refrains from freezing the liquid if it is cast underwater.

The intention seems to be that throwing the orb underwater wouldn't be considered striking the water. Striking usually refers to as one thing coming into contact with another, with the intent to hit, and when your orb 'strikes the body of water' isn't clear. It is also IN a body of water, so the orb isn't really capable of striking a body of water if it is already in one.

If the globe strikes a body of water or a liquid that is principally water (not including water-based creatures), it freezes the liquid to a depth of 6 inches over an area 30 feet square

Emphasis mine.

While getting into a bit of semantics depth is the distance from the top of something to the bottom. It would be slightly unintuitive if one could consider something 'the top of' if it started in the middle of something, such as the ocean or body of water you are in. Depth doesn't seem to apply properly in this scenario.

If it froze underwater using area and depth would be an inconvenient way to do things.

If the intention of the spell was to properly freeze while cast underwater it would, for more ease and out of a normal expectation, probably freeze in a radius or a more spherical shape. The wording of the spell seems to indicate it is made to freeze the surface of water. It is measured in area and with depth included, not a measurement and description you would use for underwater (at least not normally).

With this and the phrasing of 'strike' and with the observation that the area the spell freezes is an 'area' (with a six inch depth) it can be argued that the orb would not instantly explode with contact with water, if it were already underwater. But further on...

It would not freeze any water.

The freezing of the water is not directly related to the cold damage the spell does. With the orb not properly preforming it's second function, the freezing of water, there is no reason that the freezing would take place. Cold damage does not freeze water without DM intervention (or a certain effect says so, like with the case of some spells like Wall of Water).

Lastly...

There is no difference between casting the spell or taking the orb and throwing it.

It shatters on impact, with the same effect as the normal casting of the spell. You can also set the globe down without shattering it. After 1 minute, if the globe hasn’t already shattered, it explodes.

Emphasis mine.

Outside of a slight word difference between 'explodes' and 'shatters' the spell functions the same whether you are throwing it or casting it and throwing it instantly as apart of the spell. There is no difference between the orb shattering or exploding in a body of water. As long as the orb doesn't strike the water it shouldn't be.

Perhaps more is needed....

If designer intent was that the spell was to instantly freeze water it hit (or started in), excluding water based creatures such as elementals, then it could be that this interpretation is incorrect. I've tweeted Crawford over the issue to see what the intent is.