Choosing the right Antenna for radio chip

Yes, you will get more signal strength from a 7 dBI antenna than a 2.2dBI (specifically 4.8 dB). It solves that by radiating energy more directionally than an idea antenna that radiates evenly in all directions (0 dBI).

This increased signal strength of 4.8 dB is 10^(4.8/10) = 3 times more power. That will increase your range by about 70% in ideal conditions.

Since it is directional, you will need to point it more carefully. Specifically the linked antenna is pretty much a vertical wire. It radiates in a circle around the antenna; your receiver shouldn't be much above or below this plane.

There are many different definitions of bandwidth and engineers are used to switching back and forth between them and comparing apples and oranges to get the correct answer that they both are fruit. For example, if one says that the bandwidth of a lowpass filter is 10 kHz, then usually it means that the output power of a signal at 10 kHz is attenuated by a factor of 2 (3 dB attenuation) compared to the output power at DC. It is not the case that signals above 10 kHz are blocked entirely; if that were the intent, then the filter would have been referred to as an ideal lowpass filter with a cutoff at 10 kHz. For commonly used low-pass filters, the output power decreases at the rate of n dB per octave as the frequency increases beyond the 3 dB point, where n depends on the filter order: sharper decreases require higher-order (and thus more expensive) filters.

Similarly, if your antenna is usable from 2400 MHz to 2588 MHz, then I would hesitate at using it for signaling at a carrier frequency of 2588 MHz since the upper sideband would be attenuated considerably compared to the lower sideband. You want to be sure that the entire signal bandwidth fits comfortably within the specified range of operation.