Am I correct in saying gerund is a verbal noun?
Going home I met Joe.
Which is the gerund here?
'Going home'? Yes, I think. What is 'going'? A non-finite verb in the gerund phrase the direct object of which is 'a house', nicht war?

Am I correct in saying participle is an adjectival phrase?
'Walking horse'
Walking is a participle, a verbal adjective. 'Walking horse' is a gerund phrase as well.

Swimming in the churning river with broken legs at night is dangerous.
Let us analyze this sentence.
I give it a try and should I be wrong please rectify my mistakes.
Swimming in the river with broken legs at night is a gerund phrase, and is the subject of the sentence.
Swimming in the churning river is a gerund phrase too.
'churning river' is a verbal participle, a noun phrase.
In the churning river is a prepositional phrase.
The churning river is the direct object of the preposition, and a noun phrase.
'with broken legs' is an adverbial
'broken legs' is a verbal participle.
'is' is the finite operator, the verb.
'dangerous' is an adverbial.

let us go for a walk.
To walk is healthy.

walk and to walk are verbs in noun role, and infinitives.

The house at the end of the street.
'at the end of the street' is it an adverbial phrase or an adjectival phrase?
What is the head of the phrase? end?

in linguistics, the head is the morpheme that determines the category of a compound or the word that determines the syntactic type of the phrase of which it is a member.

'a legless chicken' is a phrase.
Let us speak about its head.
According to what the definition of head says, the head of the phrase is, in my opinion, is leg? not 'legless' because it is a compound (lexeme) and not a morpheme.
I do not really get this.
'leg' does not determine the syntactic cathegory of the phrase -- the phrase is adjectival and also noun one. 'leg' is only a noun and not an adjective, so the head of the phrase does not determine anything.
No. 'legless chicken' is not an adjectival phrase. Maybe I have got it now.

Thank you for your patience.