1) The Rambam writes in ch. 1 of Hil. Chagigah:
ומי שבא לעזרה ביום ראשון, ולא הביא עולה--לא דייו שלא עשה מצות עשה, אלא שעבר על מצות לא תעשה: שנאמר "לא ייראו פני ריקם"
If one fails to make aliya la'regel and bring a korban to the Mikdash one has violated both a mitzvas aseh as well as a lav. The Turei Even (Chagiga 7) asks: with respect to the lav of "lo yeira'u panei reikam" the prohibition is clearly defined as coming to the Mikdash emptihanded; however, with respect to the mitzvas aseh of aliya la'regel the Torah never says that anything has to be brought -- the Torah simply says three times a year to come to the Mikdash. Why according to the Rambam if one comes emptihanded has one not at least fulfilled the mitzvas aseh of aliya la'regel?
Putting aside how to answer this question (see the Minchas Chinuch's discussion at the end of parshas Re'eh), on a philosophical level I think the Rambam makes sense l'shitaso. The Rambam views Mishkan/Mikdash as simply an intrument necessary for bringing korbanos. There is no point to coming to the Mikdash just to be there. Coming to the Mikdash, like the Mikdash itself, serves a functional need to enable korbanos to be offered but is not an end in itself.
2) R' Yitzchak (Megillah 10) says one may offer korbanos in Beis Chonyo, a temple that was built in Egypt. The gemara deduces that R' Yitzchak must hold that kedusha rishona of Mikdash was temporary and after the destruction of the Mikdash it became permissable to offer korbanos anywhere. However, the gemara continues that R' Yitzchak backed out of his chiddush and even denied saying it. The sugya concludes with a machlokes Tanaim whether kedusha rishona was permanent or temporary.
Tosfos quotes Rabeinu Chaim's question [parenthetical aside: my son and I were trying to see how many Tosfos we remember that quote Rabeinu Chaim -- we are up to 4 so far. Hint: 2 in Kesubos, 1 in Kiddushin, 1 here in Megillah. I'm sure we haven't exhausted the list -- any others you know of?]: why did R' Yitzchak retract his statement when his underlying assumption is supported by at least some Tanaim?
Tosfos answers that the Tannaitic debate over kedusha rishona is simply whether korbanos can be offered at the spot of the mizbeyach even though no Mikdash stands. No Tanna, however, would allow korbanos to be brought in Beis Chonyo outside Eretz Yisrael.
If the kedushas Mikdash evaporates after its destruction, why is it that korbanos can only be offered in the place of the Mikdash? If the Mikdash no longer functions as the place of korbanos, why can they not be offered anywhere? It sounds to me like R' Chaim Kohen's approach fits better with the approach of the Ramban that sees the Mishkan/Mikdash as having inherent keduash above and beyond its functional sanctity as the place in which korbanos are offered. It is this special sanctity of the Mikdash as the place of the Shechina which remains even after the physical Mikdash is no longer present and it is this kedusha which prevents sacrifices from being offered elsewhere.
(This second idea is admittedly less convincing than the first. See the Meshech Chochma on VaYikra for a different approach to understanding Tosfos in Megillah.)