Mating types in fungi (and somatogamy)
Background:
Most fungi can reproduce sexually and asexually, and some even parasexually. Fungal reproduction is complicated!
Idiomorphs  are at the exact same locus (like alleles) but produce a completely different protein rather than a variant of the protein.
Heterothalic - also called outbreeding: male and female reproductive structures are in different individuals. Partners have opposite mating-types with compatible MAT idiomorphs
Homothalic - also called inbreeding: male and female reproductive structures in the same individual. Partners have the same mating type but both MAT idiomorphs are present in the same genome and used in mating-type switching.
Heterothallic species are said to be homogenic incompatible, heterogenic incompatibility occurs when the offspring of a heterothallic cross between different strainss shows negative heterosis. There are forms of heterogenic incompatibility that appear to act only in the sexual phase.
Note that mating type and sex are not the same as sex is determined by mating structure and mating type may have no effect on sexual structure only on the nucleic content.
Answers:
1. I am not certain, but my understanding is that a haploid spore will generate a haploid mycelium of the same mating type. Note that things can get more complicated because you can have bifactorial incompatible mating types (e.g. two unlinked mating loci)
2.  Homothallism and heterothallism are properties of the of the species
3.  I am not entirely sure I understand the question here. In Basidiomycota (Club Fungi) Mycelia of + or - mating type will fuse to form a dikaryotic (two nuclei) mycelium for the majority of the life cycle. Only under unfavorable conditions will karyogamy (the fusing of the haploid nuclei) occur to form diploid hyphae and fruiting bodies where meiosis will take place to create haploid spores with mating types which will generate the new mycelia. Part of what makes the lifecycle here difficult is remembering that diploid somatic cells will have both mating types. (Life Cycle)