Rama (O.C. 660) quotes that minhag Ashkenaz is that an aveil does not do the hakafos with his lulav during hosha'anos. Beis Yosef disagrees and wonders why an aveil should be exempt. After all, an aveil is obligated in all mitzvos, including simchas Yom Tov, which cancels aveilus, so why is hosha'anos any different? Achronim try their best to justify the minhag, but the answers are not (as the Taz and others already say) are not completely satisfying. The Bach suggests that the problem stems from the fact that an aveil cannot cut his hair, and the kohanim who did the hakafa of the mizbeiach that we are commemorating were not permitted to have long hair, peru'ei rosh. The problem is that the shiur of hair that is called too long is a growth of more than 30 days, and except for aveilus on parents, an aveil can take a haircut at the end of 30 days. Some see participating in hosha'anos as representing the tzibur, and just like an aveil cannot serve as the shat"z on days that tachanun is not recited, so too, he cannot participate here. The GR"A (quoted in M"B as well) says the exemption is based on the fact that the hakafos of the mizbeiach are a kiyum of "u'smachtem lifnei Hashem Elokeichem shivas yamim" and an aveil lacks the ability to be b'simcha (begging the question of how this is different than the mitzvah of simchas Y"T). In short: it's a difficult minhag to justify. (The Gesher haChaim paskens that an aveil can do hakafos after the tzibur as finished, and he quotes the Ya'avetz as allowing the aveil to participate on Hosha'ana Rabbah. I'm not sure how that would fit with the GR"A.)
Gesher haChaim (vol 2 ch 17) raises another fascinating question. The Kaf haChaim and others writes that although an aveil cannot participate in dancing on Simchas Torah, he is allowed to have a hakafa with the sefer Torah. If hakafos with the lulav, which are only a zeicher to the "u'smachtem" of the mikdash, are prohibited, how then are hakafos of Simchas Torah, which are part of the simchas ha'yom itself, permitted?! (Note that the Chelkas Ya'akov here prohibits participating in hakafos for this very reason.)
One theory the Gesher haChaim offers is that there is a fundamental difference between the kiyum simcha that happens in the mikdash, commemorated by hosha'anos, and other kiyumim of simcha. The simcha of the mikdash was a chovas hatzibur -- it was a communal celebration, a national rejoicing. Simchas Yom Tov celebrated by having a meal with family, Smchas Torah celebrated by dancing in shul, is a simcha of the yachid, the individual. We may come together as families to celebrate or as communities to celebrate, but the end goal in doing so is to enhance our personal kiyum of simcha. It does not transform the kiyum into a communal one. Therefore, the Rama prohibits only participating in hosha'anos but no other simcha as only hosha'anos contains a "zeicher l'mikdash" component that reminds us of this communal kiyum in the mikdash.
Even if one accepts the sevara as correct, perhaps there is room to argue. Abarbanel in Parshas VaYeilech writes that our celebration of Simchas Torah is based on the hakhel ceremony. Just like in hakhel the king would read Sefer Devarim before the assembled people, so too, on Simchas Torah we gather together and the leader of the community finishes the reading of Devarim (see Kli Chemdah in P' Zos haBracha who discusses the source for a talmid chacham being called for the final aliya without noting this Abarbanel and the parallel to melech. I am sure many Rabbis are wondering about the validity of the comparison between themselves and a melech : ) Maybe I'm wrong, but it seems to me that the hakhel ceremony, done in the makom mikdash, is a communal obligation. Everyone -- men, women, and even children -- must come and participate in hakhel. If that's what our celebration on Simchas Torah is modeled after, then the same logic that precludes an aveil from hosha'anos should apply here.
One final note: Many shuls hold a big kiddush or lunch on Simchas Torah and I was wondering if one can participate during aveilus. This post is pretty long already so I'll skip going through the shitos about whether and what simchos an aveils can/cannot participate in -- let's keep it simple: social occasions are out. I view most kiddushes as social occasions. However, it turns out that Simchas Torah lunch is an exception. Even those who hold that an aveil cannot participate in a siyum would allow an aveil to make a siyum himself -- in that case, the aveil is not a guest; he is the host. On Simchas Torah, each one of us is not a guest at the siyum haTorah -- each one of us is the host, the mesayeim. Each one of us is celebrating our personal completion of another round of learning.
And one last final, final note: I don't know how you balance "v'samachta b'chagecha" with being aware of and feeling the pain of those killed al Kiddush haShem in Eretz Yisrael. All I know is that we need lots of rachamei shamayim, lots of Torah, and we need lots of zechuyos.