Argthtjtr

The D&D 5th edition rules about the staff of the python state the following:

You can use an action to speak this staff's command word and throw the staff on the ground within 10 feet of you. The staff becomes a giant constrictor snake under your control and acts on its own initiative count. By using a bonus action to speak the command word again, you return the staff to its normal form in a space formerly occupied by the snake.

On your turn, you can mentally command the snake if it is within 60 feet of you and you aren't incapacitated. You decide what action the snake takes and where it moves during its next turn, or you can issue it a general command, such as to attack your enemies or guard a location.

If the snake is reduced to 0 hit points, it dies and reverts to its staff form. The staff then shatters and is destroyed. If the snake reverts to staff form before losing all its hit points, it regains all of them.

I have some doubts about it:

• Do you need to cast the command spell to tell the snake what to do?


• If you don't need to cast the command spell to tell the snake what do, do you spend an action to do so?


• Can you, in a single turn, return the staff to its normal form and then turn it into a snake again, thus restoring its HP to full?

The Staff of the Python has nothing to do with the Command spell.

Nowhere does it say you need to use the Command spell. The item says you need to use the command word to activate it's special properties (turning into/back from a snake). The Dungeon Master's Guide rulebook explains these (also described in the basic rules), but they are special words that are unique to each magic item that allow it to be used. If an item says it needs a command word, it will say so in the description (like it does for the Staff of the Python).

Since the description doesn't say it takes an action to control the snake, it doesn't, and you can do it freely on your turn. The snake doesn't act until its own turn, so you need to tell it what to do in advance.

Yes, you can turn the snake back into a staff, then turn it back into a snake, replenishing it's HP, in a single turn.

Note that you need to throw the staff to activate it, so you need to be next to the snake to do this. You would use a bonus action to transform the snake into the staff, an object interaction (first one on a turn is free) to pick up the staff, and then your action to throw the staff and turn it back into a snake.

The command word to activate the staff is not the Command spell

The relevant part of your quote is this:

You can use an action to speak this staff's command word [...]

The staff's command word is not the command spell, as if that were the case, it would refer to casting command using language relating to spells (such as "cast", "spell" or at least writing command in italics).

Also note that the item "requires attunement by a cleric, druid, or warlock", and many clerics, druids, and warlocks cannot have the command spell (only some archetypes can, such as a warlock with the Fiend patron, or a cleric with the Knowledge domain). It doesn't make sense that an item that only certain classes can use to require a spell that those classes can't learn (unless they have specific archetypes).

As far as I'm aware, no magic item requires the casting of a spell to function. Many items do have command words though. From the DMG, pg. 138:

... the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.

And more explicitly stated under Activating an Item (DMG, pg. 141):

Activating some magic items requires a user to do something special, such as holding the item and uttering a command word.

And again under its own section, Command Word (DMG, pg. 141):

A command word is a word or phrase that must be spoken for an item to work.

Hence the command word referred to in the item's description is the kind of command word from these above three quotes, not the spell command.

As for your other questions, it only takes your action to initially change the staff into a snake, as described in the quote at the top of my answer. Controlling the snake thereafter isn't said to use your action, so we can assume it doesn't require any action or bonus action as it doesn't say that it does, and simply says that "you can" do it:

On your turn, you can mentally command the snake if it is within 60 feet of you and you aren't incapacitated.

Also yes, the staff and therefore the snake will regain all hit points if transformed back into a staff and then into a snake again, as is implied by this part of the description:

If the snake reverts to staff form before losing all its hit points, it regains all of them.

And this can all be done on one turn, since:

[b]y using a bonus action to speak the command word again, you return the staff to its normal form in a space formerly occupied by the snake.

and then

[y]ou can use an action to speak this staff's command word [... and then t]he staff becomes a giant constrictor snake [...]

Commanding the snake doesn't require a spell

Also, the command word is not the same as the command spell

While the word "command" is used for all three things, the item description nowhere requires the user of the staff to use the command spell. The mental command to tell the snake what to do does not require an action. To address your nested questions in order:

1. 
   Do you need to cast the Command spell to tell the snake what to do? No
   


2. 
   If you don't need to cast the Command spell to tell the snake what
   do, do you spend an action to do so? No, no action, nor bonus action, is required. You give it a mental command.


3. 
   Can you, in a single turn, return the staff to its normal form and
   turn it into a snake again, thus regaining full HP? Depends on the order of operations. (see below)
   

Detailed discussion.

Command words are features of some magic items that are often discovered during attunement, or when using the identify spell on an item.

Command Word

A Command word is a word or phrase that must be spoken
for an item to work. A magic item that requires a Command word can’t
be activated in an area where sound is prevented, as in the area of
the Silence spell. (DMG)

You do not have to burn a level 1 spell slot, command, in order to use this staff. The command spell is used against humanoids to try and make them do something¹.

How to make the staff work for you

use an action to speak this staff's command word and throw the staff
on the ground within 10 feet of you. The staff becomes a Giant
Constrictor Snake under your control and acts on its own initiative
count. By using a bonus action to speak the command word again, you
return the staff to its normal form in a space formerly occupied by
the snake.

On round 1, you can transform this staff into a snake, and then with a bonus action on any round / turn, return it to its staff form. Both the action and bonus action take place on your turn. The snake has its own turn.

Using the command word to do that requires an action.

On your turn, you can mentally command the snake if it is within 60
feet of you and you aren't incapacitated. You decide what action the
snake takes and where it moves during its next turn, or you can issue
it a general command, such as to attack your enemies or guard a
location.

In this case, you are not required to use an action to do this. It's sort of like talking, or "interacting with an object." Just tell the snake what to do. The snake will do that on its turn.

Command was used in two different ways for this item's description

1. Using the command word that you find out to activate it.


2. You tell the snake what to do, which does not require an action, nor a bonus action; it just requires the PC to make a mental effort (so tell the DM what your command to the snake is).

As to your last question ... depends on the order of operations.

Can you, in a single turn, return the staff to its normal form and turn it into a snake again, thus regaining full HP?

If you use an action to activate the staff into snake form, your problem with trying to use a bonus action to stow it again is that you can't use a bonus action on the snake's turn.

Giant

Constrictor Snake under your control and acts on its own initiative
count.

The Snake gets its own turn to take actions. You just tell it what to do on its turn.

So no, you can't do it all on the first turn. To do that would require a Reaction (during the Snake's turn) not a Bonus Action (on your turn). On your next turn, you could, if you wished, use your bonus action to restore its staff form.

But what you can do is:

Round 1:

Your Turn 1: command word (action) get a snake. Mentally command it to attack.

Snake's Turn 1: attacks, also takes damage.
Round 2:

Your bonus action Turn 2: snake, become a staff!

Your action Turn 2: staff, become snake! (command word)

Mentally Turn 2: tell it what do

Snake's action Turn 2: whatever you told it to do. (At full HP).

If the snake reverts to staff form before losing all its hit points,
it regains all of them.

That all fits into the action economy as built into the game.

¹ Command spell

Command

level 1 / enchantment / Casting Time: 1 action /
Range: 60 feet / Components: V / Duration: 1 round / You speak a
one-word command to a creature you can see within range. The target
must succeed on a Wisdom saving throw or follow the command on its
next turn.