Variant calling on BAMS from different Alignment Tools

I have a question about performing variant calling on multiple BAM files that have been generated using 2 different alignment tools (for example - 'BWA-mem' and 'Novoalign' in my case).

For example - Total 4 Samples and one Bam per Sample. Sample 1 & 2 BAMs are from BWA-mem. Sample 3 & 4 BAMs are from Novoalign. And all 4 of these BAMs go through the usual best practices QC after alignment (Duplicate removal/BQSR etc.)

1. Is it recommended to use HaplotypeCaller on Sample 1, 2, 3 & 4 together (either in gVCF mode or regular mode)? (Since their BAMs were generated using different alignment tools)
2. If yes, any specific QC to perform pre-calling on BAMs (or post-calling on VCFs) to ensure the compatibility of BAMs with one another?

Shalabh