Guardo el hilo dental en el estuche para no olvidarlo cuando viajo.

Guardo is the first-person singular (yo) form of guardar in the present tense.
So it means I keep / I store / I put away (as a habitual action or something generally true):

• Yo guardo = I keep/store

In this context, guardar usually means to put away / store / keep (in a specific place).
It’s not save in the computer sense (though it can mean that too), and it’s not keep in the sense of maintain. It’s more like:

• I keep my floss in the case…

Spanish commonly uses the definite article (el/la/los/las) with everyday objects when speaking generally or habitually.
So Guardo el hilo dental… is very natural.

You can drop the article in some contexts (more like a list or very general statement), but it often sounds less natural here.

Yes: el hilo dental is masculine because the head noun is hilo (thread), which is masculine: el hilo.
Dental is just an adjective describing what kind of hilo it is.

Estuche means a case/pouch/container, often something like a small protective case. Depending on context, it could be:

• a toothbrush/floss case
• a small travel pouch
• a pen/pencil case (very common meaning)

For toiletries, some people might also say neceser (toiletry bag), but estuche is still understandable.

Because en expresses location: in/inside.

• Guardo X en Y = I store/keep X in Y

A usually marks direction or recipient (to/toward), not the location where something is kept.

Also note: a el would contract to al (unless El is a name/title).

Para + infinitive is used when the subject stays the same (I do both actions):

• Guardo… para no olvidar… = I store it… in order not to forget…

The no goes right before the infinitive: no olvidarlo.

If the subject changed, you’d typically use para que + subjunctive instead.

Lo is a direct object pronoun meaning it, and it refers to el hilo dental (the floss).

With an infinitive, Spanish commonly attaches the pronoun to the end:

• olvidar + lo → olvidarlo = to forget it

You generally cannot place it like English (para no lo olvidar is not standard).

In Spanish, after cuando, the present tense is often used for habitual/repeated actions:

• cuando viajo = when(ever) I travel

If you mean a specific future trip (not a habit), Spanish often uses subjunctive:

• cuando viaje = when I travel (in the future / next time I travel)

Both can be correct depending on meaning.

Yes, that’s another natural option, but it has a slightly different structure:

• olvidarlo = to forget it (direct object)
• olvidarme de él = to forget about it (reflexive + de)

For an object like floss, para no olvidarlo is very straightforward and common.