This ¨la¨ refers to ¨la sesión¨(session).

Se pasó toda la sesión pidiendo la palabra.

Does the phrase 'pidiendo la palabra' mean 'objecting' in a legal sense only?

You can use ¨pedir la palabra¨ in others circumstances like in a meeting,or whenever you want to express something in a group.

Could one have said 'Se la pasó haciendo objecciones todo el tiempo' also?

Absolutely correct, you can also say: Se la pasó haciendo objeciones todo el tiempo. (Objeción is with one c)

Yenny:  Ese se es la palabra...no le concedía la palabra al fiscal

Yenny, creo que te equivocas.  En este caso la palabra se se refiere al fiscal, no la palabra.  Por eso hay la palabra la. 

Debería haber dicho:

Yenny:  Ese se es la palabra al fiscal...no le concedía la palabra al fiscal..

O lo que es igual que :

No se la (la palabra) concedía.

¡Gracias!

Se la pasó pidiendo la palabra

y

No se le olvidará más poner la señal

Porqué usa "se" en los dos ejemplos?  Significaría algo diferente sin "se"?

En el segundo frase, a que se refiere "le"?

Gracias.  Fue una lección excelente.

In the first sentence the se belongs to the verb pasarse.  The la is the situation, or what was going on, the thing.  La is used like this in Spanish for a lot of things. 

¿Cómo la ves?  What do you think?

Que te la pases bien.  Have a good time.  (just one example of have a good time)

pasarse can have many meanings depending on context, here I would translate as he spent the whole time either objecting or asking permision to speak, not 100% sure about pidiendo la palabra here.

In the second example, the le refers to the defendant.  The business owner who didn’t put the warning sign “caution wet floor“ out. 

You could say the same thing like this: No olvidará más poner...

If it was more than one person the two forms would look like this:

1. No se les olvidará más

2. No olvidarán más

If I was talking to you with tú:

1. No se te olvidará más

2. No olvidarás más

The thing is though, using olvidar in the reflexive form like this is common and takes a little bit of the edge off in some situations.  Like it’s not as acusatory, you can place blame eslewhere.

Think of them as things that are happening to you or someone kind of indirectly.

Olvidé las llaves.  I forgot the keys.

Se me olvidaron las llaves.  I forgot the keys.  (the keys forgot themselves on me) not my fault.

Oye, ¿donde estabas?

Uyyy, se me olvidó completamente! 

[warning: This explanation may cause dizziness, nausea, headache,  and exfolliative dermatitis. Use under grammarnerd supervision only.] 

As I see it, there are two kinds of se; let's call them se azul and se rojo.

SE has a very specific and limited use: In this sentence... 

     I give him the book / Le doy el libro.

... LE is the indirect object of 'dar' and it means 'to him'.

But if I want to say:

     I give it to him / Le lo doy --> Se lo doy

... that is 'le' changes to 'se' in this environment.  It is usually explained as a sound change: Spanish "doesn't like the way le lo sounds" so it changes the le to se.

The same thing happens with the plural form, les, when it precedes a direct object pronoun.

     les + lo --> se lo / it to them.

So se azul can mean: to it, to him, to her, to them, to you (formal) singular or plural) when it is followed by a direct object pronoun such as lo (but also me, te, la, nos, os, los and las)

It's much easier to keep this straight in your head if you remember some examples:

     Se lo juro / I swear it (to you)

     Se lo prometo / I promise you (it)

     ¡Dígaselo! / Tell him (it)!

The order here is rule-based:  indirect object precedes direct.  Se azul is an indirect object pronoun and it always comes first. 

Then there is SE ROJO.  This is the se that gets tacked on to the end of verbs in the dictionary. This is a sorta, kinda, reflexive se in that it refers back to or reflects the subject of the verb.  In the sentence:

     No se le olvidará más poner la señal /He won't forget to put out the sign any more. (Putting the sign out will not forget itself to him any more)

...se refers back to its grammatical subject, 'poner la señal.'

...(le refers back to an unstated logical, not grammatical subject: 'a él')

This particular use of se rojo is, in my opinion, the trickiest and is best learned by examples like:

     Se me olvidó la cartera / I forgot my wallet (My wallet forgot itself to me)

     Se le cayó el plato / He dropped the plate (The plate fell to/on him)

Se rojo isn't breaking rules of order (se lo prometo!) because it's not an object pronoun. 

Now se rojo has a bunch of uses, including reflexive (se lava las manos), reciprocal (se conocen/they are acquainted with each other) and impersonal (aquí se habla inglés / they speak English here) just to name a few.)

If you want to know more about this se, look in a good grammar book under 'pronominal verbs'.