It should be quite easy for the DAQ-Card to read a 300mV signal at a sampling rate of 1kHz.
I can't explain why you are seeing peaks of "2.5V and above 10V" - this sounds quite wrong - in fact most DAQ-Cards are not even capable of directly measuring anything greater then 10V.
Most DAQ cards will measure using a default range of -5V to +5V (instrumentation amplifier gain of x1) unless you specify otherwise.
In traditional DAQ programming, you specify your expected maximum signal range and LabVIEW/DAQ chooses a suitable board gain to cover at least your signal range. Board gain values are typically 0.5, 1.0, 0.5, 0.2, 0.1, 0.05, etc, etc, although not all boards have as full a range as some others do (it varies depending on cost/specification) - for example, the NI-PCI-6025E low cost board only has gain/ranges of 0.5 (+/-10.0 V), 1.0 (+/-5.0 V), 0.1 (+/- 500 mV) and 0.01 (+/- 50mV).
So to answer your specific questions...
i) No the problem you are seeing is not due to your lowish signal.
ii) Your card can amplify the voltage prior to the DAC (digital to analogue converter) - although you need to program this as you deem appropriate.
iii) Normally you don't need to amplify the signal yourself (except in special circumstances).
Final advice - you should always try to use the default "differential" measuring mode in order to minimise noise - and check to make sure that your signal source is ground referenced to your PC/DAQ board by some route.
Mark H.