Why do the IVs change based on level?

Solution 1:

You have to use all the information available to you, including the in-game appraisal system. Learn what the phrases your team leader uses mean, and use that information to help you narrow down the possibilities.

Example

Take this Weedle as an example

If you enter in the information from this screen into the IV Calculator at Poké Assistant, you'll end up with 14 possibilities.

If you use the in-game appraisal, the team leader indicates that HP is the best stat, and that it's either 13 or 14. If you enter in that information, you'll only be left with 6 possibilities.

However, the team leader also said that the Pokémon was "decent" (team Valor), so you know the Pokémon must be between 51.1% - 64.4% perfect. This means that you can exclude the last four items that Poké Assistant listed, leaving you with only two possibilities.

From there you can further narrow the possibilities down by powering up the Pokémon, and entering the new values into the calculator. Using a second calculator can also help you narrow it down.

At any rate, the evaluation is based on the Pokémon level, not the trainer level. You can estimate the level of a Pokémon, by looking at the stardust required to power it up. In my example above, the Weedle must be either level 13 or 14, based on the stardust required to power it up.

To answer the question directly, the IVs don't change based on level. What an IV calculator is showing you, is all the possible IV combinations that would produce a Pokémon with the stats you entered.

In the above example, a level 13 Weedle with 5/0/14 IVs, and a level 13 Weedle with 0/10/14 IVs, both result in a level 13 Weedle with 131 CP and 45 HP. The only difference would be the IV distribution, and the percent perfection.

Solution 2:

IVs stay the same throughout levels on the same Pokemon, and are randomly generated when you obtain them. Pokemon levels are also random when caught but can be increased by powering up.

You can powering up a Pokemon, to a max level which increases based on your trainer level. The arc over your Pokemon shows how far it is from reaching the max level. Whenever your trainer levels up, the arc becomes less filled because the maximum increased.

For example, you can have a perfect IV (15/15/15) Pikachu that is lvl 10, which has a lower CP than the 9/2/6 Pikachu, which is lvl 30. The lower-IV Pikachu is still stronger than the perfect IVs but lower level Pikachu, but will be weaker if the lvl 10 perfect Pikachu becomes powered up to the same level.

Pokemon at the same level will differ in stats/CP based on their IVs. Pokemon with better IVs will become stronger per level compared to inferior Pokemon.

Solution 3:

Pokemon Stats

IVs never change, they are the "genetic makeup" of the individual Pokemon. They are created when the Pokemon spawns. You can see this if multiple people catch the same Pokemon spawn, they will have the same IVs but random CP. They are used in combination with your level (which is the random part of spawns) to calculate your CP and HP.
Note: IVs (which are Attack, Damage and Stamina) and Level are unavailable as is the calculation used.

Trainer Level

This puts limitation on what level you can train your Pokemon to and that's all.

IV Calculator Websites/Apps

These things take your estimated level using stardust required to reach the next level, together with your CP and HP and try to do the reverse of this hidden calculation to estimate the unknowns (IVs).

So why are there multiple possibilities?

Because sometimes multiple combinations of Attack, Damage, Stamina and an estimated level can lead to the same CP, your Pokemon IVs can't be accurately calculated. The same way if you have a calculation

 10 = x + y

You can say that x=5 and y=5, but you could also say x=1 and y=9 or many other possibilities.

How do I get around this?

You need to narrow the possible IV/Level of your Pokemon:-

1. Do a calculation
2. Note the possible IV/Level options
3. Level your Pokemon
4. Note the possible IV/Level options that are common between two tests
5. If there are more than 1 options remaining, go to step 3.

This works because you then have information on how your CP/HP is calculated at different levels. More data = more accuracy. The downside to this is that you might need to waste candies and startdust to find out you have a 2.2% Pokemon.

Alternatively

The other option is to understand the rating system. Each team leader has different phrases which give you a hint as to which possibility is the right one, you can Google for spreadsheets that tell you exactly what each of the leader statements indicate.