Q: How much of this can be done by using the budget reconciliation process to avoid filibusters?
If an item is in the president's budget proposal, the budget reconciliation process may be used. There are, by my count, 236 line items in the budget proposal; all of which could be done, in a single bill (with concurrence of both houses), without a Senate filibuster.
Q: Would all such legislation have to be wrapped up in a package as part of a single yearly budget reconciliation bill?
Only the budget items will be "wrapped up" in a single bill, sometimes called an omnibus reconciliation bill. The process of "wrapping up" is the reconciliation process.
The purpose of the reconciliation process is to allow Congress to use an expedited procedure when considering legislation that would bring existing spending, revenue, and debt limit laws into compliance with current fiscal priorities established in the annual budget resolution. In adopting a budget resolution, Congress is agreeing upon budgetary goals for the upcoming fiscal year (as well as for a period of at least four additional outyears). In some cases, for these goals to be achieved, Congress must enact legislation that alters current revenue, direct spending, or debt limit laws. In this situation, Congress seeks to reconcile existing law with its current priorities.
Q: Issues like immigration and voting rights seem to me like they have very little to do with government spending, but if they have some budget implication, is that enough?
No, such items are excluded by the Byrd Rule.
The Byrd rule was first adopted in 1985 in response to concerns that committees were including recommendations in their reconciliation submissions that were extraneous to achieving the budgetary goals established in the budget resolution. The Byrd rule generally prohibits the inclusion of material considered extraneous to the purpose of a reconciliation bill.
Q: Does it depend completely on how sympathetic the senate's parliamentarian is?
No, the Senate parliamentarian is considered an independent (non-partisan) advisor to the presiding officer of the Senate.
Q: Are such rulings bound by precedent, or can the current parliamentarian just do whatever they want?
The role of the parliamentary staff is advisory, and the Presiding Officer may overrule the advice of the parliamentarian. In practice, this is rare; the most recent example of a Vice President (as President of the Senate) overruling the parliamentarian was Nelson Rockefeller in 1975. That ruling was extremely controversial, to such an extent that the leaders of both parties immediately met and agreed that they did not want this precedent to stand, so the next week the Senate altered the rule under consideration via standard procedure.
Q: Can anyone break this down into smaller pieces and discuss which pieces probably would qualify for the reconciliation process and which wouldn't?
The smallest possible piece, as given above, is that if it's in the president's budget, it qualifies for reconciliation; otherwise, it doesn't.
Note that the House has completed its omnibus bill "pursuant to title II of S. Con. Res. 14" (priced at $1,748.72 billion for FY 2022) and is available for examination. When the Senate completes its bill, the two will be reconciled to produce the final bill, which bill requires only a majority vote in each house.
See also, The Budget Reconciliation Process: Stages of Consideration