For instance, how would I say "walking" in the sentence "Walking across the beach, I listened to the waves"?

Also, how would I say "walking" if I was talking about my "walking abilities"?

1) Promenante laŭ la strando, mi aŭskultis la ondojn.

2) "iraj kapabloj" / 'irkapabloj" / "kapabloj iri".

I think it somewhat surprising that in the Esperanto translation of the Bible, when it refers to Jesus walking on water the verb iri is used. This doesn't seem to me to be precise and makes it less of a miracle as to how he achieved this, whereas actually walking on water does seem miraculous.

Marŝi would have been a better choice.

Again if someone was wounded and I wished to know if he could walk, I would naturally enquire 'ĉu vi povas marŝi'.

Eh, you picked a surprisingly hard topic. Firstly there are several "walkings" in English. Secondly what is "to walk" in Esperanto and the third is what kind of beach.

The three -ing forms of English

gerund

The Latin grammar term gerundium refers to an infinitive that acts like a noun. For this meaning a simple infinitive will do in Esperanto.
 

• Dancing in a good hobby.
  
• Das Tanzen ist ein gutes Hobby.
  
• Danci estas bona hobio/ŝatokupo.

participle

Present participle is a verb form that denotes on-going, non-completed action. It is often used in expression where the subject does something else at that moment. For this Esperanto uses participles.
 

• Are you watching tele?
  
• Ich suche einen tanzenden Frosch.
  
• Mi serĉas dancantan ranon.

deverbal noun

As the name implies these are nouns that have been derived from verbs. For this Esperanto often uses nouns with the -ado ending.
 

• Loud shouting makes me angry.
  
• Unter einer Drehung versteht man in der Geometrie…
  
• Laŭta kriado kolerigas min.

your example

Your example clearly falls into the second category: while doing something, the subject did something else.

To walk

I once wrote an article about the difficult of expressing the act of walking in Esperanto. You can take a look at my article (in Finnish, but try Google translate, no guarantee).

The verb iri is the general one, to go, which is usually understood to mean on foot. I take walking on a beach in your example to mean walking without a goal, just for fresh air or something along that line, so I would choose promeni.

The beach

Esperanto has the Fundamental bordo as well as later additions in the form of the French-based plaĝo and the German-based strando. The Fundamental bordo is clear, a piece of land touching any water, let it be sea, lake or river. The other two are less clear. According to PIV strando is an even bordo with sand for swimming and taking sun. By the same source plaĝo is a strando prepared for people on holiday. But I bet that many speakers happily mix these – or whether the definitions in PIV are even accepted by all. Since you are walking on this beach, let us take strando (beach with sand) as Nornen already did.

Putting the pieces together
 

• Promenante laŭ la strando mi aŭskultis la ondojn.