Even brand name 9 V alkalines sometimes leak, but not often and usually only when left in the instrument even when dead for a long time. However the amount of fluid is often small, so the damage can be limited. I have not seen NiMH Cell leak, but similar made NiCd sometimes do.
Whether rechargeable is worth it, depends on the usage. Standard NiMH tend to have too much leakage to be useful in a low current application. If you need to replace the batteries often (e.g. more than every 3 months) it might be worth the rechargeable.
It depends on the meter, how well the operate with a lower voltage when the battery is near empty. Some are more picky than others. I know of some calculators the could not reliably work with NiMHs (AAA) because only worked down to about 1.25 V / cell. Similar things might happen in a DMM.