All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Our answer to your question will be summarized in the following:
1- First of all, Abu Haneefah is of the view that order is not necessary if the missed prayers are more than one day and one night, i.e. if they exceed five prayers and not six prayers as you mentioned. This is what Ibn Qudaamah attributed to Abu Haneefah in Al-Mughni. He attributed it to the Maaliki school as well, saying: “Maalik and Abu Haneefah said that sequence is not an obligation for prayers of more than one day and one night, because anything beyond that is burdensome and may lead to repetition. So it is not necessary, as with the sequence in making up for missed days of fasting from Ramadan.”
This is also what Imaam An-Nawawi ascribed to them. He said in his book Al-Majmoo’ about the obligation of sequence according to them: “Abu Haneefah and Maalik said that it [sequence] is an obligation as long as the missed prayers do not exceed those of one day and one night.”
What we have come across in the literature of the Maaliki School is that a distinction is only made between a large or small amount of missed prayers for the prayers missed and the prayer whose time is current [That is, it is not obligatory to pray the missed prayers before the current prayer if the prayers missed were more than a day and night's worth]. They say, however, that it is an obligation to pray the prayers missed in sequence in relation to each other, without distinction between a large or small amount.
2- In the example you mentioned about missing Thuhr prayer one day and ‘Asr prayer another day without knowing which one was first, sequence is not necessary according to one of the two narrations in the Hanbali School, because it is impossible to observe sequence in this case; Ibn Qudaamah said in Al-Mughni: “If a person missed Thuhr prayer one day and ‘Asr another day and does not know which came first, then there are two quotes about this: one of them is that it suffices to guess from the circumstances which one was missed first and then make up for them in that order... The second quote is that he should pray Thuhr then ‘Asr without investigating which prayer was missed first. Muhanna quoted Ahmad as saying this. The rationale is that investigation means to look for signs of evidence, and there is no sign of evidence in this case, so the sequence which is put forth in Sharee'ah is instead given consideration...”
3- The opinion of senior Hanbali scholars, as it is well-known, is that it is an obligation to observe the sequence even if there are many missed prayers, and some of them are of the view that it is recommended only. Al-Mardaawi mentioned both opinions in Al-Insaaf, saying: “His statement "in sequence whether they are a few or many": this is the general view and the view of the majority of the scholars of the Hanbali School, contrary to the other three Schools of jurisprudence. Imaam Ahmad is also reported as saying that the sequence is not an obligation. Al-Mubhij reads: 'The sequence is recommended and this is the preferred opinion in Al-Faa’iq.' Ibn Rajab said in Sharh Al-Bukhari: 'Some scholars of the Hanbali school have decisively asserted this opinion and favored it, saying that because Imaam Ahmad was extremely cautious, he would take the safer side in issues of controversy. Otherwise, the opinion that it is obligatory to make up many years of missed prayers for the sake of a single prayer is hardly based upon strong evidence.' He said: 'A certain scholar of our Hanbali School informed me that he saw the Prophet in a dream and asked him about whether Ash-Shaafi'i's opinion and Ahmad's is preferred in these matters of differing. He said: What I understood from this is that he [the Prophet] pointed to Ash-Shaafi’i's opinion as being preferred.' It has been said that sequence is an obligation only for five missed prayers. Al-Qaadhi chose this view in one instance. Al-Furoo’ reads: 'It is probable that sequence is obligatory, but it does not affect the validity (of prayer). There are other issues that are analogous to this one.”
To summarize the quote from Al-Insaaf, sequence is not an obligation according to one of the two quotes from Ahmad; the author of Al-Mubhij, Abu Al-Faraj Ash-Shiraazi, stated this and it is the preferred opinion of the author of Al-Faa'iq, Ahmad ibn Al-Hasan ibn Qudaamah Ad-Dimashqi Al-Hanbali, known as Ibn Qaadhi Al-Jabal. Ibn Rajab, author of Fat-h Al-Baari Sharh Saheeh Al-Bukhari (not Ibn Hajar who authored the more celebrated Fat-h Al-Baari), seemed to prefer this opinion, saying: "The obligation of making up many years of missed prayers for the sake of a single prayer is a great burden that is not in line with the principles of the merciful religion of Islam."”
4- The last part of the quote we cited from Al-Insaaf, about sequence not affecting the validity (of the prayer), means that if one does not observe sequence, then his prayer is still valid, and not observing sequence does not render the prayer invalid. In his commentary on Al-Kaafi, Shaykh Ibn ‘Uthaymeen said about the sequence not being a condition for the validity of the prayer: "It is not a condition that if not fulfilled the prayer is not valid. That is to say that it is not like the condition of Ghusl (major ablution) for Janaabah (ritual impurity), or the condition of Wudhoo' (minor ablution) for minor impurity. The difference is that it [sequence] has to do with the manner [of the act of worship] and not the act of worship itself. The act of worship is complete except for the order. For this reason, it [sequence] is excusable for forgetfulness, ignorance and fear of the time for prayer running out, and fear of missing the congregational prayer, according to one opinion. If it is said that it [sequence] is an unfulfilled obligation and can be made up by repeating it, we say that it being an obligation is questionable. And even according to the view that it is an obligation, it is not like the obligation of saying the Tashahhud in prayer for example. And it cannot be made up by repeating it because Allaah did not command us to perform one prayer twice. So, evidence is required for the obligation of repeating it.”
The opinion that performing the missed prayers in order is an obligation but not a condition for the validity of the prayer is the adopted view in the Maaliki School. Minah Al-Jaleel reads: “The adopted view is that praying the missed prayers in order is an obligation and not a condition.”
The view of the Hanbali school is that it is a condition for the validity (of the prayer). After mentioning two quotes [from Ahmad] about the obligation of observing sequence and that the first one is that it is an obligation, Ibn Muflih said in Al-Mubdi’: “According to the first quote, the order is a condition for the validity, so if unfulfilled, the prayer is invalid, as with Rukoo’ (bowing) and Sujood (prostration). According to Al-Furoo', 'There is a likely probability that order is obligatory but it does not affect the validity (of the prayer).'”
This is the answer to your question, "...does this mean if one doesn't make them up in order, then in their opinion the prayer is invalid?"
Allaah Knows best.