Learn Hardcore Spanish (Spain): Quiero dejar de perder tiempo y seguir estudiando español. - I want to stop wasting time and keep studying Spanish.

Dejo el libro en la mesa = I leave the book on the table.
Déjame en paz = Leave me alone.
Dejar perder would mean “to let [something] be lost,” not “to stop losing it.”

Should it be "perder tiempo" or "perder el tiempo"? Which sounds more natural?

The idiomatic way to say “waste time” is perder el tiempo. Many speakers would prefer:

Quiero dejar de perder el tiempo...

Perder tiempo (no article) is also used and understood, often with the sense of “to lose time” (e.g., to be delayed) or when speaking more generally/quantitatively:

No quiero perder tiempo con trámites.
Perdimos tiempo por el tráfico.

In your sentence, either works; adding el makes the “waste time” idea explicit and is slightly more idiomatic.

Can I use "parar de" instead of "dejar de"?

Yes. Parar de + infinitive is common in Spain and often fairly informal:

Quiero parar de perder el tiempo...

Nuance:

Dejar de is the default for “to stop doing.”
Parar de often emphasizes ceasing an ongoing or repetitive action and is very common in negatives: No para de hablar = “He won’t stop talking.”

Why is it "seguir estudiando" and not "seguir estudiar"?

With seguir and continuar, Spanish uses the gerund (-ando/-iendo) to express “to keep/continue doing”:

Seguir/continuar + gerundio: Sigo estudiando español = I keep studying Spanish.

You do not use seguir + infinitive in this meaning. Related negative pattern: seguir + sin + infinitive = “to keep not doing”:

Sigo sin estudiar = I still haven’t started studying.

Do I need to repeat "quiero" before the second verb? Is the coordination correct?

No need to repeat it. Quiero governs two coordinated phrases:

Quiero [dejar de perder (el) tiempo] y [seguir estudiando español].

Repeating quiero is possible for emphasis, but not necessary: Quiero dejar de... y quiero seguir...

Should the "y" change to "e" before "seguir"?

No. Spanish changes y to e only before words that start with the “ee” sound (spelled i- or hi-): padres e hijos, agua e hielo, amigos e invitados.

Since seguir starts with an “s” sound, you keep y: ... y seguir ....

Why is "español" lowercase and why no article before it?

In Spanish, names of languages are written in lowercase: español, inglés, francés.
After verbs like hablar, saber, aprender, estudiar, you usually omit the article: Estudio español. You use the article when speaking about the language as a thing in general or when it’s the subject: El español se habla en muchos países.

Could I say "Yo quiero..." or is "yo" unnecessary?

Spanish normally omits subject pronouns because the verb ending shows the subject. Quiero already implies “I want.” Use yo only for emphasis or contrast:

Yo quiero dejar de perder el tiempo (as opposed to someone else).

When do I use "querer + infinitive" versus "querer que + subjunctive"?

Same subject → querer + infinitive: Quiero dejar de... y seguir... (I want to stop and to continue).
Different subjects → querer que + subjunctive: Quiero que (tú) dejes de perder el tiempo y (que) sigas estudiando.

Could I say "seguir con el español" instead of "seguir estudiando español"?

Yes:

Quiero seguir con el español = I want to carry on with Spanish (more general).
Quiero seguir estudiando español = I want to continue studying Spanish (specifically the activity of studying).

Both are natural; the gerund version is a bit more precise about the action.

If I replace "español" with a pronoun, where does it go?

You can place the direct-object pronoun either:

Attached to the gerund: Quiero seguir estudiándolo.
Before the conjugated verb: Lo quiero seguir estudiando.

Do not say Quiero lo seguir estudiando.

Is "dejar de perder" an odd stack of verbs? Is it really idiomatic?

It’s fine and idiomatic. Dejar de must be followed by an infinitive, and here that infinitive is perder. Alternatives:

Dejar de perder el tiempo (very common).
Dejar de malgastar el tiempo (a bit more formal).
Colloquial: Dejar de hacer el tonto / dejar de tonterías.

Could I use "para" instead of "y" to show purpose?

Yes, with a small nuance:

... dejar de perder el tiempo y seguir estudiando español. = two separate goals.
... dejar de perder el tiempo para seguir estudiando español. = “stop wasting time in order to keep studying Spanish” (explicit purpose).

Both are correct; choose based on the intended relationship.

Any pronunciation tips for tricky parts of this sentence (Spain)?

Quiero: the qu is a hard “k” sound; the u is silent → [ˈkje.ɾo].
dejar: j is a throaty sound (like German “Bach”): [deˈxaɾ].
seguir: gu before i gives a hard “g”; the u is silent → [seˈɣiɾ].
y (and): pronounced like English “ee” [i].
español: the ñ is a palatal “ny” sound

Is "quiero" too direct? Would another verb sound softer or more polite?

For statements about yourself, Quiero... is perfectly normal in Spain. To sound softer or more tentative, you can use:

Me gustaría dejar de... y seguir... = I’d like to...
Querría/Quisiera dejar de... y seguir... = I would like to... (more formal/polite).

How would I say this in another tense?

Near future (plan): Voy a dejar de perder el tiempo y voy a seguir estudiando español.
Simple future: Dejaré de perder el tiempo y seguiré estudiando español.
Past intention (ongoing): Quería dejar de perder el tiempo y seguir estudiando español.
Past decision (completed): Decidí dejar de perder el tiempo y seguir estudiando español.