Are the Pali words "cause" ("hetu'), "condition" ("paccaya"), "nutriment" ("ahara") and "root" ("mula") synonyms?

Paccaya/hetu/āhāra = Whole Condition (Lead&Supporting Condition) in paṭiccasamuppāda.

Mūla = Lead Condition/Root Condition in paṭiccasamuppāda.

Mūla = "Root" in pāli language. Life is similar to a tree, lobha&dosa&moha&whole unwholesome are the seed & root of life. When you want to kill this tree you have to destroy whole seed&root. Because if you destroy just the top of tree, new trees can grow up again and again by the left seeds&roots. This is the reason which pali often use mūla as Lead Condition. See the quote sutta below for more explanation.

HetuPaccaya is used by sāriputta-mahāsāvaka in paṭṭhāna-abhidhammapiṭaka because he have to avoid to conflict mūla with Sutta Pitaka Vol 4 : Sutta. Ma. Mū. {285.1} (English translation). if he use mūla in this context instead of hetu, all other unwholesome saṅkhāra can't be mūla, so paṭṭhāna will conflict with that sutta. That's the reason why sāriputta chose hetu instead of mūla for paṭṭhāna.

Paccaya/hetu/āhāra = Whole Condition (Lead&Supporting Condition)

Playing as lead condition in paṭiccasamuppāda

Tasmātihānanda eseva hetu etaṃ nidānaṃ esa samudayo esa paccayo upādānassa yadidaṃ taṇhā. --Sutta Pitaka Vol 2 : Sutta. Tī. Ma. {58.3} / (English translation)

Playing as supporting condition in paṭiccasamuppāda

tasmātihānanda eseva hetu etaṃ nidānaṃ esa samudayo esa paccayo vedanāya yadidaṃ phasso. --Sutta Pitaka Vol 2 : Sutta. Tī. Ma. {58.3} / (English translation)

Kavaḷīkāro āhāro oḷāriko vā sukhumo vā phasso dutiyo manosañcetanā tatiyā viññāṇaṃ catutthaṃ . ime kho bhikkhave cattāro āhārā bhūtānaṃ vā sattānaṃ ṭhitiyā sambhavesīnaṃ vā anuggahāyāti. --Sutta Pitaka Vol 8 : Sutta. Saṃ. Ni. [31] / (English translation)

Mula = Lead Condition/Root Condition

{79.7} Katamo ca sīha pariyāyo yena maṃ pariyāyena sammā vadamāno vadeyya tapassī samaṇo gotamo tapassitāya dhammaṃ deseti tena ca sāvake vinetīti . tapanīyāhaṃ
sīha pāpake akusale dhamme vadāmi kāyaduccaritaṃ
vacīduccaritaṃ manoduccaritaṃ yassa kho sīha tapanīyā
pāpakā akusalā Dhammā pahīnā ucchinnamūlā tālāvatthukatā anabhāvaṃ katā āyatiṃ anuppādadhammā tamahaṃ tapassīti
vadāmi tathāgatassa kho sīha tapanīyā pāpakā akusalā
dhammā pahīnā ucchinnamūlā tālāvatthukatā anabhāvaṃ katā
āyatiṃ anuppādadhammā ayaṃ kho sīha pariyāyo yena maṃ pariyāyena sammā vadamāno vadeyya tapassī samaṇo gotamo tapassitāya dhammaṃ deseti tena ca sāvake vinetīti.

{79.8} Katamo ca sīha pariyāyo yena maṃ pariyāyena sammā vadamāno vadeyya apagabbho samaṇo gotamo apagabbhatāya
dhammaṃ deseti tena ca sāvake vinetīti . yassa kho sīha āyatiṃ gabbhaseyyā punabbhavābhinibbatti pahīnā ucchinnamūlā
tālāvatthukatā anabhāvaṃ katā āyatiṃ anuppādadhammā tamahaṃ
apagabbhoti vadāmi tathāgatassa kho sīha āyatiṃ gabbhaseyyā punabbhavābhinibbatti pahīnā ucchinnamūlā tālāvatthukatā anabhāvaṃ katā āyatiṃ anuppādadhammā ayaṃ kho sīha pariyāyo yena maṃ
pariyāyena sammā vadamāno vadeyya apagabbho samaṇo gotamo apagabbhatāya dhammaṃ deseti tena ca sāvake vinetīti. -- Sutta Pitaka Vol 4 : Sutta. Ma. Mū. {285.1} / (English translation)

The Pali words "cause" ("hetu') is NOT a synonym of "condition" ("paccaya"). To explain this, let me take the example in paticca samuppada (PS). The PS cycle that describes the “cause and effect” in Buddha Dhamma starts as: “avijja paccaya sankhara, sankhara paccaya vinnana, vinnana paccaya nama rupa, nama rupa paccaya salayatana,….”.

One may tend to think that “avijja paccaya sankhara” means “avijja causes sankhara” or “ignorance causes one to acts that generate bad kamma”, but it is not so. All of us ordinay people have avijja, but that does not mean that we always act accordingly. Most of the time we act appropriately or morally. However, if avijja is there, it is likely that at times one will act with avijja and do inappropriate or immoral things.

Thus PS does not refer to a “direct link”. Any effect must have a cause. But there can be possible causes without leading to any effects. Otherwise, Nibbana would not be possible. For causes to bring about corresponding effects, suitable conditions must be present. That is what ‘paccaya’ means.

Similarly the Pali words "nutriment" ("ahara") is NOT a synonym of "root" ("mula"). In Pali language, “mula” is a root, so that “mūlika héthu” means “root causes”. To explain this, let me take the example of a tree. The roots (mūlika hetu) are critical for the tree’s survival, but the “patta” also plays an important role for the tree’s growth. The “patta” are essential for the tree’s survival, just as the roots of the tree are. One could kill a tree simply by a process called “girdling” where those phloem cells are removed. Girdling, also called ring-barking is the complete removal of a strip of bark. from around the entire circumference of either a branch or trunk of a woody plant. Girdling results in the death of the area above the girdle over time. A branch completely girdled will fail and when the main trunk of a tree is girdled, the entire tree will die.

So, the tree’s survival depends on not only its roots, but also its “patta” containing those critical phloem cells. Thus, CONDITIONS are as important as ROOT CAUSES, and both these taken together is "nutriment" ("ahara") for the tree.