How does the cyclopropyl group influence conjugation and aromaticity?
About the molecule that you remember cited in your textbook, there's a study in ref.1. According to these authors:




  The delocalization of the C1-C2 and Cl-C3 electrons of spiro[2.4]hepta-4,6-diene (1a) and partial rehybridization at C1, C2, and C3
  result in a contributing structure which is perhaps best represented as a $\pi$ complex (lb) of cyclopentadienylidene and ethylene.


From NMR studies they concluded that: 


  The comparison of the predicted downfield shift
  of approximately 6.6 ppm for the complex and the
  observed shift of approximately 1.0 ppm leads to the
  the conclusion that the $\pi$ complex may contribute as
  much as 15 % to the total electronic structure of 1.


The most recent reference I have found is from 1997(ref.2). They studied these structures:



  In the series of the investigated compounds 1-5, spi- ro[2.4]hepta-4,6-diene (1) shows the strongest  influence of cyclopropyl  conjugation on molecular structure in the group of doubly  unsaturated   cyclopropyl  hydrocarbons


The conjugation properties of cyclopropane are subject of debate and other references don't support this. You can find a blog about this on the royal society of chemistry website(http://my.rsc.org/blogs/84/1009).

I have found another paper in ref.3 and according to this:


  In one chemical attempt to test the transmission
  of conjugation by cyclopropane rings, Cannon, et al.were unable to alkylate or acylate the methyl group in diethyl 2-methylcyclopropane-l, 1-dicarboxylate and concluded that transmission of conjugative effects in the transition state for carbanion formation are small compared with analogous, unsaturated compounds.


Also:


  Trachtenberg and Odian concluded from their review
  of the various conflicting reports that a cyclopropane
  ring is incapable of transmitting conjugation in the
  ground state but may be able to do so in some excited
  states. It is likely that, during chemical reactions,
  such transmission occurs only in a transition state,
  i.e., after bond breaking of the ring has commenced.


From what I have read, even assuming a conjugate effect, this would be much less evident than a C-C double bond.

About your questions:
Is there always a conjugation between a cyclopropyl group and an adjacent double bond? In other words, do cyclopropane's "bent" bonds always have extensive π-character and can, therefore, act as an alkene?
 Partially, the effect is not comparable to a true double bond. This hypothesized conjugation effect was found to be stronger in piro[2.4]hepta‐4,6‐diene(1) compared to the other molecules studied in ref.2(molecules 2 to 6 in the second picture).

1a. Are electronic structures 1 and 1a; 2 and 2a; 3 and 3a similar, and to what extent?
Considering that the supposed conjugation contribution of the cyclopropane ring is much less than a C-C double bond, I think they differ substantially. Unfortunately, I can't find any spectra or other additional data supporting this.

Can we use Hückel's 4n+2 and 4n rules for cyclopropyl-containing monocyclic rings? If not, how to estimate aromaticity/anti-aromaticity in such system?The Huckel rule is not valid in these complex cases. The Hückel's rule is based on calculations using the Hückel method and therefore has a limited validity. Here you need advanced MO calculations and as you have seen from all the references mentioned in such systems there's a lot of debate. 

References:
1) Aromaticity via cyclopropyl conjugation. Electronic structure of spiro[2.4]hepta-4,6-diene, J. Am. Chem. Soc.197092154736-4738 https://pubs.acs.org/doi/abs/10.1021/ja00718a047

2) Structural Aspects of Cyclopropyl Conjugation: Experimental Studies and Ab Initio Calculations.  https://doi.org/10.1002/jlac.199719971007

3)Transmission of conjugation by the cyclopropane ring, Stewart JM, Pagenkopf GK, The Journal of Organic Chemistry, 1969. https://pubs.acs.org/doi/pdf/10.1021/jo00838a003?rand=vimssm6d