Angular House Conjunctions

If a planet is within five degrees of an angle, but resides in the next sign, does it still count?

Yes, it still counts. What matters most in aspects is staying within a certain orb of degrees. We don't just discard the next sign or even previous sign. All this means is the planet differing in sign from the angle, is adding another layer of expression or another theme in how the angle manifests.

Ascendant (Self, externalization, first impression.)
Out of sign planet conjunct here, makes one have more layers to their overall ego/self-expression. For example, one may wonder why an angry Aries Rising is so tender and emotional (has Moon Pisces in late degree conjunct Aries Rising at early degree).

Descendant (Others, reception of how we behave, interaction.)
Out of sign planet conjunct this, adds layers of how they may or may not easily "switch modes" in how they interact with other people. Some people may enjoy one's Gemini Descendant tendency for adult humor, but be astonished at how sensitive and empathetic the person can be about another person's trauma triggered by their humor (Mercury Cancer in early degree conjunct Gemini DSC. at late degree).

Imum Coeli/I.C. (Childhood, old age, private life, inner self as a response to those who condition us or care for us.)
Out of sign planet conjunct here, means another layer of how they feel about their past and how they presently manage their lifestyle. Someone wants to leave behind an unsupportive hometown, with Sagittarius I.C., but they also surprise their hometown by coming back with a successful business idea that both morally spites and materially supports them, with Mars Capricorn in early degree conjunct late degree Sagittarius I.C. (This is an example of motives we develop over time, as the I.C. and its ruler represent how we are conditioned.)

Midheaven/M.C. (Reputation, public image, career or status through life, outer self as in who we choose and grow to be.)
Out of sign planet conjunct this angle, is adding a layer of how one adapts and aspires to their highest potential. Someone may feel conflicted by the strong need for recognition with Midheaven Leo, the self-confliction caused by a humble and passive Moon Cancer in late degree conjunct Midheaven Leo in early degree. Such a person slowly learns that as long as their vision benefits the respective people they wish to appeal to, then they truly are meant to be held in the high regard they always crave and successfully build themselves up to be.

Any out-of-sign planet within maximum generous orb of 10-degrees away or towards conjuncting an angle will basically modify the meaning of a usual same-sign conjunction. I say this orb, as such wide conjunctions/oppositions are included in default setting for Astrodienst charts. But obviously, the closer the orb, the more significance and clearer influence there is.

The same logic applies for all aspects, but narrower orbs are necessary for trine, sextile, quincunx. Conjunctions, oppositions, and squares to the angles are practically axial alignments that indicate heavy focus in the themes involved and more obvious manifestation of the planetary energies in one's life.

If a fixed star is within two degrees of an angle, but resides in the next sign, does it still count?

Fixed stars are very specific focal points (due to distance, staying in same position, etc.). Since you are comparing an Angle's Zodiac degree by itself to only a fixed star of a differing Zodiac constellation, then it is ideal to stick to the nearest fixed star in the same Zodiac constellation as the angle. But it can be as simple as trying to wrap your mind around how a person relates to, or how their life matches the out-of-sign conjunction to a fixed star.

Thank you for the very thorough and informative response !
It clears up the odd placement here and there, that I felt really ought to count, but decided otherwise.
With all this in mind, I have a lot of corrections to make.