It's complicated.

To begin with, yes, you can cast it on yourself.

The target line is "one humanoid," with no further restrictions in the text of the spell. If you are a humanoid, legally, you can Dominate yourself.

What happens next is ambiguous.

The spell is ambiguous on several key points:

1. It is clear that the target entering the Dominated state and receiving its first order are part of the same action. But what happens first? Does the subject "become dominated" and then "receive its first order," or the other way around?

2. Can the target still think in ways that don't require an action? Or are they in a completely will-less state?

The argument for permitting it goes like this:

You cast Dominate Person on yourself, with a command something like "act the way you want to." You are now dominated, and single-mindedly performing the task of... Acting exactly like you're not dominated.

The key point in favor of this is that there is no complexity limit on commands:

If you and the subject have a common language, you can generally force the subject to perform as you desire, within the limits of its abilities.

Doing whatever you want is certainly within your abilities, therefore you profit.

It's a little cheesy, but the "be yourself" loophole isn't without precedent in genre fiction.

Argument for not permitting it (kill it with fire):

You cast dominate on yourself. You enter a dominated state, and wait for orders. Unfortunately, you no longer have the will to order yourself around, so you wait in a will-less state ("to the exclusion of all other activities except those necessary for day-to-day survival (such as sleeping, eating, and so forth).") for the duration of the spell.

Argument for not permitting it (except at a cost):

You cast dominate on yourself, and give yourself a command (but not "be yourself," because what does that even mean?). You then enter the will-less state of domination, and are unable to change the command once given.

Appendices

I. Wishful Legalism

The "Act as you want" command can backfire terribly - think about all the occasions in which a responsible, non-psychotic adult shouldn't do or say the things he happens to want at the moment - postponing gratification, acting tactfully and respecting social bounds all goes against doing what you really want... An attentive DM could make wonders with a PC in a fugue state for a few days, acting "as he wants". Different wording can backfire just as well, for example: "do what you should be doing" can cause the dominated PC to immediately cast Dominate Person on herself again! :)

And then, when the DM narrates some psychotic action you look him dead in the eye and say "I don't want to do that." At that point the DM is either violating the rules of the spell, or dictating what your character does/does not want. Either is bad.

Remember: Unlike with alcohol, you still have your full capability to predict the outcome of your actions, and use that to determine what you actually want.

But suppose your DM does say they'll permit this type of action, and then tries to weasel out of it by twisting your intent. In this case they might get you once or twice, but they have started a war they can't win. The command can be arbitrarily long, so you simply have to sit down, write something that you feel is ironclad, and keep iterating until the DM fails to screw you (you may want to do this in a secure location during downtime).

For example, the next step in the escalation would be something like this: "act as though you hadn't been dominated, except don't dominate yourself again (for the same purpose as the casting of this dominate)."

A smart DM will just flat out say "no" and pick one of the other interpretations (or declare it a house rule).

II. Saving Throws

Dominate Person allows its targets a number of saving throws. Do you need to worry about rolling too high, and accidentally breaking your own Dominate?

As it turns out, the answer is "no." You are always allowed to voluntarily forfeit a saving throw:

Voluntarily Giving up a Saving Throw

A creature can voluntarily forgo a saving throw and willingly accept a spell's result. Even a character with a special resistance to magic can suppress this quality.

Your questions:

1. Can a character under the effects of Geas cast Remove Curse on herself? YES.

2. Can this be prevented by saying "Bring me 100gp from the offering plate every day and don't cast remove curse on yourself?" NO.

3. How does this work? SEE BELOW:

Question 1:

There is nothing in the spell description stating that the creature is magically compelled to do, or to not do, anything, even disobeying the command you have given. What there is is a consequence for disobeying, which it explicitly states is possible:

While the creature is charmed by you, it takes 5d10 psychic damage each time it acts in a manner directly counter to your instructions, but no more than once each day.

So YES they can cast Remove Curse on themselves. or in fact do anything else.

Question 2:

The geas spell description (PHB p.244) states that

You place a magical command on a creature that you can see within range, forcing it to carry out some service or refrain from some action or course of activity as you decide

This means that you can tell them to do something or tell them not to do something, but not both. Also it is singular, you can refer to only a single course of action. So you can't command them to not cast Remove Curse on themselves (or anything else) as well as the primary command.

So NO you can't prevent them from casting Remove curse on themselves in addition to the primary command.

Question 3:

I am going to rephrase the "How does this work?" question to something more specific: At what point does the target of a geas know it has had a geas cast on it?

The spell description states:

If the creature can understand you, it must succeed on a Wisdom saving throw or become charmed by you for the duration.

and that

A creature that can't understand you is unaffected by the spell.

This means that they have to hear and understand your verbal command to be able to follow it, there is no "telepathic" magical transference of meaning. This means the creature effected knows at least that someone has given it a command it can hear in a language it can understand at the time of casting. As stated earlier there is no magical compulsion to follow the command.

Do they know they have had a spell cast on them? The Rules answers 2016 state, using suggestion (PHB p.279) as an example, gives strong guidance:

You’re aware that a spell is affecting you if it has a perceptible effect or if its text says you’re aware of it (see PH, 204, under “Targets”). Most spells are obvious. For example, fireball burns you, cure wounds heals you, and command forces you to suddenly do something you didn’t intend. Certain spells are more subtle, yet you become aware of the spell at a time specified in the spell’s description. Charm person and detect thoughts are examples of such spells.

Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there.

Breaking this down you’re aware that a spell is affecting you if:

• it has a perceptible effect
• if its text says you’re aware of it at some point

You might miss that a spell is affecting you if:

• the spell is subtle
• you failed to notice the spellcaster casting the spell

Applying this to Geas:

• it does not have a perceptible effect at the point of casting and
• the description does not explicitly state you are aware of it.
• The damage done by the spell is psychic, in the mind of the target and that only happens at the point you go against the command and The spell has only a Verbal component, no handwaving or bits of fleece so it is subtle (a velvet wrapped hammer springs to mind)

So it is down to whether the character notices the spell being cast as to whether they know that a spell has been cast on them, up until they take damage and at that point it is whether they recognise the spell effect for what it is.

The PHB p203 describes a Verbal component of a spell as:

Most spells require the chanting of mystic words. The words themselves aren’t the source of the spell’s power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion.

This means that the words of the command given are not the only part of the Verbal component, at the very least the words have to be said at a particular pitch and resonance etc. So it will be down to the DM to set the difficulty for the target, or any other witness, to notice and recognise that a spell has been cast.

The target has to be aware of the command so no perception roll is required so as a DM I personally would base it on an Int(Arcana) roll (or Wis(Arcana) if you, like I do, use the alternate rules on skill characteristics with a different take on the information provided by a success). I'd suggest a difficulty of around 13 depending on the circumstances.