You are absolutely right Andres.
I do not know what the "flashing display" indicates. Is it a flag on the 29c, or an error diagnostic?
Norman, I personally would recommend just replacing your old NiCad batteries with new NiCads-- it's a simple easy and cheap repair, sure-fire and guaranteed.
Alkalines could rupture if used in conjunction with a charger/power pack. So if you ever need to use alkalines in a calc, it is my habit to take a piece of cork and stopper the charge port. Just a way of reminding me not to be stupid.
Alkalines/dry cells/primary-type ZnCl batteries all put out 1.5v nominal PER CELL, while NiCads and NiMH are 1.2v/cell. Not a great deal of difference, but my personal rule is that you should not act as if alkalines and NiCads are equivalent if there will be more than 3 in series (an excess of > .9v). Most devices have regulation circuitry, true: but most regulation circuitry consists of converting excess voltage to heat, not great for reliable small devices.
Truthfully, I would be skittish about alkalines in the 25c in particular, mainly because this is VERY EARLY C-MOS, and I think it is more picky about the voltages it tolerates. I would try to keep it to NiCads.
I do not know about the 29c's circuitry, but would guess it is somewhat more robust and tolerant than the 25c. Just my personal impression. Nevertheless, Andres is right to warn you to NEVER run it from the power-pack without the rechargable batteries in place (they act as a "soft" buffer to the power from the pack).
There are many places that can rebuild a battery pack for you if you have a bad one. Luckily, you can even buy solder-tabbed AA cells on the Net, if you want to try it yourself. The packs are usually a combination of three or so AA NiCad cells, on most calcs. It is three on the 45, for instance. Opening up the battery pack carefully is probably the hardest part of the job. You can do it!
Don't solder directly onto a NiCad UNLESS it has solder tabs to solder to. If you ruin the venting system under the cap by melting it, the NiCad is never going to charge well, and might (if overcharged) burst. Katie Wasserman pointed out in an earlier post that Digi-Key sells some pretabbed NiCads, and I think that is likely an excellent and reliable source.
If you are wondering about NiMH, I have YET to see REAL decent data that NiMH, when charged by a NiCad-based charger with a NiCad based method, delivers more power overall or lasts more cycles. It has much less of a "memory" problem, as you have heard, but charging it is an exothermic process (produces heat) where NiCads are endothermic, and NiMH loses its charge as it sits idle at a 30-40% higher rate than NiCad will (all batteries lose charge when sitting, but NiCad is STILL better than NiMH in this regard, and I think this has some virtue in a calculator).
If NiMH is charged in a NiMH charger, you DO get more power capacity, and you DO get more cycles, but since HP chargers weren't originally designed for NiMH, the way to use them is to convert your calc battery pack to allow you to change batteries in it easily, then keepp a set of NiMH in your charger, and a set in the calc, and switch them out RATHER THAN trying to charge them IN THE CALC.
By "converting your pack to allow you to change batteries easily", I mean you can, for instance, use the NiMH cells with an industrial cap (a bit shorter than the button-topped consumer variety), and use a spring-metal contact in each side of the pack, and "seal" the pack after each change with scotch tape. It works, and even someday, can be easily reversed to a plain ol' NiCad pack again.
Be picky. Power is the only thing a calculator needs to do its work-- and on such a diet, the best you can do to keep your calculator healthy is provide it with the RIGHT stuff...