Syntax: 'My tea's gone cold' uses the present perfect tense ('has gone') to indicate an action completed in the recent past with present relevance (the tea is now cold). 'I'm wondering why' is a present continuous clause acting as the main clause, showing an ongoing mental state.
Morphology: 'Tea's' is a contraction of 'tea has'. 'Wondering' is the present participle of the verb 'wonder'. 'Got' is the simple past of 'get', used informally here. 'At all' is an adverbial phrase used for emphasis in negative or interrogative contexts.

Syntax: 'And even if I could, it'd all be gray' is a conditional sentence (second conditional) expressing a hypothetical situation and its imagined result. 'It'd' is a contraction for 'it would'. 'It reminds me that...' introduces a noun clause ('that it's not so bad') as the object of the verb 'reminds'.
Morphology: 'Could' is the past tense modal verb used for hypothetical ability. 'Gray' is used as an adjective describing a state. 'Reminds' is third person singular present tense. 'Bad' is an adjective in its base form after the copula 'be' ('it's').

Syntax: 'I missed the bus and there'll be hell today' uses coordination with 'and' to connect two independent clauses. 'There'll be' is a contraction of 'there will be', forming a future existential statement. 'I'm late for work again' uses the present simple to describe a current state resulting from a past action.
Morphology: 'Missed' is the simple past tense of 'miss'. 'There'll' is a contraction of 'there will'. 'Hell' is a noun used idiomatically. 'Late' is an adjective. 'Again' is an adverb of frequency.

Syntax: 'And I want to thank you for giving me...' uses an infinitive phrase ('to thank you') as the direct object of 'want'. 'For giving me...' is a prepositional phrase where the object of the preposition 'for' is a gerund phrase ('giving me the best day'). 'Oh, just to be with you is having...' uses an infinitive phrase ('to be with you') as the subject of the sentence, with 'is having' as the verb phrase.
Morphology: 'Want' is a stative verb in present simple. 'Thank' is a verb in its base form after 'to'. 'Giving' is the gerund/present participle form of 'give'. 'Best' is the superlative form of the adjective 'good'.

Syntax: 'Push the door, I'm home at last' uses an imperative-like structure ('Push the door') to describe an action, followed by a declarative statement. 'And I'm soaking through and through' uses the present continuous to describe a current state resulting from a past action (walking in the rain). 'Then you handed me a towel' uses simple past for a sequential narrative action.
Morphology: 'Push' is a verb in base form, used descriptively. 'Soaking' is the present participle used as an adjective. 'Through and through' is a repeated preposition for emphasis. 'Handed' is the simple past of 'hand'.