Q: A Hydran fleet has cut off the Lyran supply route into the captured (and now former) Hydran capital. The Klingons, however, have a valid supply path, as well as both PDUs and a BATS at the former Hydran capital. Are the Lyrans in supply by (410.4) since they're with a friendly base?
A: There would have to be a homeless supply line (410.5) established or an expeditionary fleet (411.7) established to leave the hex and operate. The forces, per (410.25) would be in supply as long as they are in the same hex as the friendly base.
Q: By the definition of (413.1), a Supply Grid is a network of bases and/or planets that contains a friendly capital hex or off-map area. Also by definition, a Partial Supply Grid (413.4) is any other grid that does not contain the capital or off-map area.
Here is where it gets confusing. The Federation's supply path has been cut off from the capital in 2908 to the off-map area. By definition, the off-map area is not a Partial Supply Grid, but in fact is a (regular) Supply Grid. There are rules (413.44) concerning EPs in a regular grid and a partial grid, but (technically) never address the question of two separate regular grids. I am not trying to claim that EPs generated in the off-map area are magically available to the Capital Grid, but, technically, (413.44) doesn't cover two "regular" grids (one Off-Map Grid and one Capital Grid) so there is (technically) no rule about how to move those EPs back and forth once the connection is rebuilt. So, can I use the (413.44) rules to move any EPs stockpiled at the Off-Map Grid to the Capital Grid when the two are reconnected? Also, can the Off-Map Grid use deficit spending since it is not a partial grid?
A: You have found a valid gray spot in the F&E2K rules which we have fixed in the F&E2KX rules, and everyone is grateful that you noticed this just in time for F&E2KX. By definition (413.1) there can be a maximum of two Main Supply Grids: a Capital Grid and an Off-Map Grid. Each can do anything that a Main Supply Grid can do (since they both are Main Supply Grids) except of course that the "main shipyard" is only in one of them. EPs in either grid can be used, by and in that grid, for valid uses such as ship repairs (420.0), PDU self-generation (433.42), base repair (420.6), conversions at non-capital starbases (433.0), etc. If two main grids are separated and then joined, the EPs again flow freely between them, so you can certainly use (413.44).
As to your second question, only the Capital Grid can use Advanced Deficit Spending (447.0) in PO.
Q: Can I build a Klingon B9 fast battleship by direct buy, or only by paying for a series of die rolls?
A: It depends on what rules are used in your game or scenario. If using (436.0), you pay for a series of die rolls, and if using (525.1) in AO, you can buy it outright.
Q: Capitals can make one major (4+ EP) conversion at a starbase, except that the Federation and Klingons can make two such major conversions. Are they both made at the same starbase, or does each take a separate starbase?
A: Each takes a separate starbase, so the Federation capital (with three starbases) can make a total of three conversions (433.11) of which two (433.12) can be major conversions. Rule (450.1) in PO allows any empire to build additional conversion facilities, and of course, more starbases could be built.