Is there an external world beyond sense impressions at all?

Yes there is an external reality. The biggest mistake in modern philosophy--beginning with Descartes--was to cast doubt concerning an external reality. Modernists generally posit our inner states of affairs (e.g., thoughts, perception, etc.) as the only indubitable objects of experience.

Here's the problem: How can "sense impressions" be experienced, if to be 'experienced' means it's be 'seen', 'touched', 'tasted', or 'heard' of?

(1) If we experience them [impressions] by means of the senses, and since an external reality was doubted because it was an object of the senses, then sense impressions must too be doubted on similar grounds. This leads to complete skepticism, which suffers the regress of skeptical attitudes.

(2) If, on the other hand, these 'impressions' are not experienced at all, then no knowledge can be had of such impressions, leading to skepticism again. Similarly, if we experience them through our minds and not through our senses, and if minds consist of the same inner states affairs, then circular reasoning ensues.

(3) We can try and posit another impression to account for the previous impression i.e., having a sense impression of the sense impression. But we will face the same problems encountered in (1) and (2), and have to posit another impression ad infinitum. An infinite series of sense impressions implies that there is no endpoint to the layers of impressions, such that there is no final impression and thus no experience of anything at all.

Conclusion: We DO experience or are aware of something. Our inner states of affairs cannot be objects of those experiences. Therefore, we must be experiencing an outer state of affair or an external reality.