davening after 4th hour

Rabbi Yosef Avraham Heller, the Rosh Kollel of Crown Heights, Brooklyn and former member of the Beis Din there, wrote a essay explaining the Halachic justification for davening after Chatzos, published in "Kobetz Beis Chayenu" 11 Nissan 5760 pg. 28. The crunch of the explanation is as follows:

The Gemora (Brochos 26a) states that, "He may go on praying [Shachris] the whole day. But up to midday he is given the reward of saying the Tefillah in its proper time; thereafter he is given the reward of saying Tefillah, but not of saying Tefillah in its proper time." Although the mainstream view in Rishonim is that of the Rashba that after Chatzos is only considered tashlumim if he accidently missed the time, the Perisha in OC Siman 89 cites an alternative view that even after Chatzos is still considered the time of davening Shachris but only that one would not receive the reward for davening on time. Rabbi Heller continues to back up this view based on many Rishonim that maintain that the time for Shachris is the entire day. Although this Prisha argues with the Mechaber and Rama, Rabbi Heller suggest that since it concurs with the view of many Rishonim, Chassidim relied on his ruling in order to have adequate kavana in their davening.

There are also several letters of the Lubavitcher Rebbe on the topic collected in "Sharey Halacha U'Minhag" OC Chelek 1 pg. 111. The Rebbe brings the halacha (OC 98:2 and Rambam Hilchos Tefila 4:15-16) that proper kavanah during davening is integral to the davening, and without it the teffila is nullified. He asserts that therefore proper preparation for davening takes precedence over being exact in the time of davening.

However, it must be stressed that all the above is only when one is involved in davening or it's preparation from before zman tefila. There is no justification to begin davening after chatzos.

In terms of an opinion that as long as one begins before zman tefilla they may continue even after, I heard that this comes from Tosfos in Brochos 7a "Sh'ilmaley". The Gemora relates that Biliam tried to find the exact second Hashem was angry to curse the Jews, and Tosfos asks what he could have said in that short moment. In their second answer Tosfos says that as long as he would have started at a time of Divine anger the curse would work even after. Similarly, the "zman tefilla" follows when one begins even if he finishes after.

https://judaism.stackexchange.com/questions/22003/halachic-justification-to-not-pray-the-amidah-before-zman-tfilah

http://moshiachtalk.tripod.com/bikores/bikores00.htm

http://moshiachtalk.tripod.com/bikores/bikores00.htm

rebbe as moshiach

https://judaism.stackexchange.com/questions/38256/did-the-late-lubavicher-rebbe-consider-himself-a-to-be-a-prophet-and-or-the-mash

learning Chumash, saying Tehillim, and Learning Tanya daily

R' Zev Dov Slonim compiled a booklet that collates various sources that talk about the advantage of learning Chumash, saying Tehillim, and Learning Tanya daily. The booklet is also included in the Chitas sefer, published by Kehot. The footnotes on Sefer Haminhagim's entry on Chitas discusses this as well.

Here are some of them, see sources inside for more:

• The Shibolei HaLeket (Inyan Tefilla #44), brings from Rashi that the early Chassidim would learn for an hour after prayer. He connects this to the idea brought in the Talmud (Kiddushin 30A) , that one should always split his his learning into three. A third Mikrah, a third Mishna, and a third Talmud.
  Rabbi Slonim points out that in Shulchan Aruch HaRav (Hilchot Talmud Torah, Chapter 2, Halacha 1), the Baal HaTanya says that explanations of the Written Torah fall under the heading of "Mishna" and Kabbalah falls under the catagory of "Talmud".
  Hence, learning Chitas daily covers Mikrah (Tehillim, Chumash), Mishna (Rashi on Chumash), and Talmud (Tanya).

https://judaism.stackexchange.com/questions/17761/the-value-of-the-daily-study-of-chitas-chumash-tehilim-tanya/48064

Rebbe's ship manifest. 1941

Rav Avigdor Miller on The Lubavitcher as Moshiach Q: If our generation is the עקבתא דמשיחא, the generation of the footsteps of Moshiach, why should our generation be the lowest of all generations? A: Now, I don't know if this is the generation of עקבתא דמשיחא, that I can't tell you, but let's say it is. The answer is — you know what it says:  כי יראה כי אזלת יד — When Hakodosh Boruch Hu will see that all the power is gone from us, ואפס עצור ועזוב, that there is nothing left, ואמר אי אלקימו, and we will see there is nobody to help us, so that’s when Hakodosh Boruch Hu will come to help. That's what the Chofetz Chaim said; he said, “Please Ribono Shel Olam. It’s getting late and we’re going lost.” He said, “Please come quickly and help us.” That's a reason why Moshiach should come — just because we’re the lowest of all generations. Of course, it would much better if Moshiach should come because we’re so good. However, nobody knows what can happen yet — don’t be so sure that this is the end; it can chalilah get much worse. I want to tell you a word from the old Lubavitcher Rebbe — I'm not a Lubavitcher, but the old Lubavitcher Rebbe said something. In his time, they published a newspaper called הקריאה והקדושה — it was the old Lubavitcher Rebbe. And there was a certain year when they said Moshiach is going to come this year. He put himself out on a limb. "Moshiach will come this year,” he said. It was years ago – the old Rebbe, not this one. And the Lubavitcher? They were organizing, making pronouncements: “This year! This year!” Everyone else was ridiculing it. And the year passed by and nothing happened. So I offered him a consolation; my little cheap consolation I offered to them. I said, “The Lubavitcher Rebbe is right; he was right. He himself redeemed many Jews from going lost; he was a Moshiach for many Jews.” Who says your job is to worry about the Moshiach for the whole Klal Yisroel? What about yourself? You're living only once in this world, and your job is to be a Moshiach to rescue  at least yourself. And if somebody comes and rescues you like the old Lubavitcher Rebbe did, he’s also a Moshiach. The old Lubavitcher Rebbe founded a yeshiva in Rochester, a yeshiva in Buffalo, a yeshiva in Boston. He founded yeshivos wherever he went. In those days they didn't have such a thing. It was before Torah U'Mesorah. How many children, how many Jewish boys and girls was he a Moshiach for? And therefore Moshiach came; in a certain sense it came. You have to worry only about the whole Klal Yisroel?! You have to worry about each one of us! So Moshiach did come! And therefore, although it's a virtue: אחכה לו בכל יום שיבוא – Everybody should hope every day that Moshiach should come, but in the meantime get busy and redeem yourself. You hear that? Don't wait. There’s a lot to accomplish. TAPE # 593