I must adress that this confusion is mainly due to the fact that English words do not always change from verb to noun, but they do change in Spanish.

So, answering your first question:

No, you won't find verbs acting like their associated noun.

Indeed you can see verbs acting like nouns, but, as guifa said, that's used with other meaning (the act of doing). Like in

Smoking is bad = Fumar es malo.

Secondly; it is actually very usual to use "ir a trabajar", because it is supposed that you only "work" at one place: "your work". Of course you do many more things in life, but "working at your job" can only be done "at work". That's why we usually say "Vo a trabajar". It's the same as saying "Voy a comer" = I'm going to eat. It's like that's the important information, the place where you do it is understood.

However, if you want to make a difference between the verb and the noun, you can:

Voy al trabajo

That's "I'm going to work(noun)", to your workplace.

You might say that "trabajo" is also a verb, but (besides workplace), "trabajo" means "I work", not the verb "to work". Verbs themselves do not act like their associated nouns, but some forms might coincide with the noun, by chance.

Finally, the conclusion is that you can use nouns, but they are not the same word. Take into account that verbs have to end in "ar", "er", "ir", whereas nouns do not.

In your examples, the nouns are:

I am going to play/I am going to a play

Voy a jugar / tocar / interpretar // Voy a una obra.

I store things there/He bought it at the store

Almaceno/Guardo cosas ahí // Lo compró en el almacén / en la tienda.

I plan on doing that/It is a plan/We made plans

Planeo hacer eso / Es un plan / Hicimos planes.

I talk/We had a talk

Yo hablo / Tuvimos una charla

I must adress that this confusion is mainly due to the fact that English words do not always change from verb to noun, but they do change in Spanish.

So, answering your first question:

No, you won't find verbs acting like their associated noun.

Indeed you can see verbs acting like nouns, but, as guifa said, that's used with other meaning (the act of doing). Like in

Smoking is bad = Fumar es malo.

Secondly; it is actually very usual to use "ir a trabajar", because it is supposed that you only "work" at one place: "your work". Of course you do many more things in life, but "working at your job" can only be done "at work". That's why we usually say "Voy a trabajar". It's the same as saying "Voy a comer" = I'm going to eat. It's like that's the important information, the place where you do it is understood.

However, if you want to make a difference between the verb and the noun, you can:

Voy al trabajo

That's "I'm going to work(noun)", to your workplace.

You might say that "trabajo" is also a verb, but (besides workplace), "trabajo" means "I work", not the verb "to work". Verbs themselves do not act like their associated nouns, but some forms might coincide with the noun, by chance.

Finally, the conclusion is that you can use nouns, but they are not the same word. Take into account that verbs have to end in "ar", "er", "ir", whereas nouns do not.

In your examples, the nouns are:

I am going to play/I am going to a play

Voy a jugar / tocar / interpretar // Voy a una obra.

I store things there/He bought it at the store

Almaceno/Guardo cosas ahí // Lo compró en el almacén / en la tienda.

I plan on doing that/It is a plan/We made plans

Planeo hacer eso / Es un plan / Hicimos planes.

I talk/We had a talk

Yo hablo / Tuvimos una charla.